\section{Introduction}
Rado asked in 1966 (see Problem P531 in \cite{rado1966abstract}) if it is possible to extend the concept of matroids to infinite
without loosing
duality and minors. Based on the works of Higgs (see \cite{higgs1969matroids}) and
Oxley  (see \cite{oxley1978infinite} and \cite{oxley1992infinite}) Bruhn, Diestel, Kriesell, Pendavingh and Wollan
settled Rado's problem affirmatively in \cite{bruhn2013axioms} by finding a
set of cryptomorphic axioms for infinite matroids, generalising the usual independent set-, bases-, circuit-, closure- and
rank-axioms for finite mastoids. Higgs named originally these structures B-matroids to distinguish from the original concept.
Later this terminology vanished and in the context of infinite combinatorics B-matroids are referred as matroids  and the term
`finite matroid' is used to
differentiate.

An $ M=(E, \mathcal{I}) $ is a matroid  if $ \mathcal{I}\subseteq \mathcal{P}(E) $ with
\begin{enumerate}
[label=(\arabic*)]
\item\label{item axiom1} $ \varnothing\in  \mathcal{I} $;
\item\label{item axiom2} $ \mathcal{I} $ is downward closed;
\item\label{item axiom3} For every $ I,J\in \mathcal{I} $ where  $J $ is $ \subseteq $-maximal in $ \mathcal{I} $ but $ I $ is
not, there exists an $  e\in J\setminus I $ such that
$ I+e\in \mathcal{I} $;
\item\label{item axiom4} For every $ X\subseteq E $, any $ I\in \mathcal{I}\cap
\mathcal{P}(X)  $ can be extended to a $ \subseteq $-maximal element of
$ \mathcal{I}\cap \mathcal{P}(X) $.
\end{enumerate}

If $ E $ is finite, then \ref{item axiom4} is redundant and \ref{item axiom1}-\ref{item axiom3} is one of the usual
axiomatizations
of finite matroids. One can show that every dependent set in an infinite matroid contains a minimal dependent set which is called
a circuit. Before Rado's program
was settled, a more restrictive axiom was used as a replacement of \ref{item axiom4}:

\begin{enumerate}[label={(4')}]
\item\label{item axiom4'} If all the finite subsets of an $ X\subseteq E $ are in $ \mathcal{I} $, then $ X\in \mathcal{I} $.
\end{enumerate}

The implication \ref{item axiom4'}$ \Longrightarrow $\ref{item axiom4} follows directly from Zorn's lemma thus  axioms
\ref{item axiom1}, \ref{item axiom2}, \ref{item axiom3} and \ref{item axiom4'}  describe a subclass  $ \mathfrak{F} $ of the
the matroids. This $ \mathfrak{F} $ consists of the matroids having  only finite circuits and called the class of \emph{finitary}
matroids. Class $ \mathfrak{F} $ is closed under several important operations like direct sums and taking minors but not under
taking
duals which was the main motivation of Rado's program for looking for a more general matroid concept. The class $
\mathfrak{F}^{*} $ of the duals of the matroids in $ \mathfrak{F} $
consists of the \emph{cofinitary} matroids, i.e. matroid whose all cocircuits are finite. In order to being closed under all
matroid operations we need, we work with the class $
\mathfrak{F}\oplus \mathfrak{F}^{*} $  of matroids having only finitary and cofinitary components, equivalently that are the
direct sum of a finitary and cofinitary matroid.

Matroid union is a fundamental concept in the theory of finite matroids. For a finite family $ (M_i: i\leq n) $ of matroids on a
common finite edge set $ E $ one can define a matroid $ \bigvee_{i\leq n}M_i $ on $ E $ by letting $ I\subseteq E $ be
independent in $ \bigvee_{i\leq n}M_i $ if  $ I=\bigcup_{i\leq n}I_i $ for suitable $ I_i\in \mathcal{I}_{M_i} $ (see
\cite{edmonds1968matroid}). This
phenomenon fails for  infinite families of finitary matroids. Indeed, let $ E $ be uncountable and let $ M_i $ be the $ 1
$-uniform
matroid on $ E $ for $ i\in \mathbb{N} $. Then exactly the countable subsets of $ E $ would be independent in
$ \bigvee_{i\in \mathbb{N}}M_i $ and hence there would be no maximal independent set contradicting
\ref{item axiom4}.\footnote{For a finite family of finitary matroids the union operation  results in a matroid (Proposition 4.1
in \cite{aigner2018intersection}).} Even so,
Bowler and Carmesin observed (see section 3 in \cite{bowler2015matroid}) that the rank formula in the Matroid Partition
Theorem by Edmonds and Fulkerson (Theorem 13.3.1 in \cite{frank2011connections}), namely:
\[ r\left( \bigvee_{i\leq n}M_i \right) =\max_{I_i\in \mathcal{I}_{M_i}}\left|\bigcup_{i\leq n}I_i\right|=\min_{E=E_p\sqcup
E_c}\left|E_c\right|+\sum_{i\leq n}r_{M_i}(E_p),\]
can be interpreted in infinite setting via the complementary slackness conditions. In the minimax formula above there is equality
for the family $ (I_i: i\leq n) $  and partition $ E=E_p\sqcup E_c $ iff
\begin{itemize}
\item $ I_i $ is independent in $ M_i $,
\item $\bigcup_{i\leq n} I_i\supseteq E_c $,
\item $ I_i\cap E_p $ spans $ E_p $ in $ M_i $ for every $ i $,
\item  $ I_i\cap I_j\cap E_p=\varnothing $ for $ i\neq j$.
\end{itemize}
 Bowler and Carmesin conjectured that for
every family $\mathcal{M}:= (M_i: i\in\Theta) $  of matroids on a common edge set $ E $ there is a family $ (I_i: i\in \Theta) $
and partition $
E=E_p\sqcup E_c $ satisfying
the conditions above. To explain the name ``Packing/Covering Conjecture'' let us provide an alternative formulation.  A
\emph{packing} for $ \mathcal{M} $ is a system $ (S_i: i\in\Theta ) $ of pairwise disjoint subsets of $ E $ where $ S_i $ is
spanning in $ M_i $. Similarly, a \emph{covering} for $ \mathcal{M} $  is a  system $ (I_i: i\in\Theta ) $ with
$\bigcup_{i\in\Theta} I_i=E $ where $ I_i $ is independent in $ M_i $.

\begin{conj}[Packing/Covering, Conjecture  1.3 in \cite{bowler2015matroid} ]\label{conj: Pack/Cov}
For every family $ (M_i: i\in \Theta) $ of matroids on a common edge set $ E $  there is a partition $E=E_p \sqcup E_c$ in such a
way that $ (M_i \upharpoonright E_p: i\in \Theta) $ admits a packing and $ (M_i.  E_c: i\in \Theta) $ admits a covering.
\end{conj}

We shall prove the following special case of the Pacing/Covering Conjecture \ref{conj: Pack/Cov}:

\begin{restatable}{thm}{PC}\label{t: main result0}
  For every family $ (M_i: i\in \Theta) $ of matroids on a common countable edge set $ E $ where $ M_i \in
  \mathfrak{F}\oplus \mathfrak{F}^{*} $,  there is a partition $E=E_p \sqcup E_c$ such
  that $ (M_i \upharpoonright E_p: i\in \Theta) $ admits a packing and $ (M_i.  E_c: i\in \Theta) $ admits a covering.
\end{restatable}

It is worth to mention that packings and coverings have a crucial role in other problems as well. For example if $ (M_i: i\in\Theta)
$ is as
in Theorem
\ref{t: main result0} and admits
both a packing and a covering, then there is a partition
$ E=\bigsqcup_{i\in \Theta}B_i $ where $ B_i $ is a base of $ M_i $ (see \cite{erde2019base}). Maybe surprisingly, the failure
of the analogous statement
for arbitrary
matroids  is consistent with set theory ZFC (Theorem 1.5 of \cite{erde2019base}) which might raise some scepticism about the
provability of  Conjecture
\ref{conj: Pack/Cov} for general matroids.

The Packing/Covering Conjecture \ref{conj: Pack/Cov} is closely related to the Matroid Intersection Conjecture which has
been one of the central open problems in the theory of infinite matroids:

\begin{conj}[Matroid Intersection Conjecture by Nash-Williams, \cite{aharoni1998intersection}]\label{MIC}
If $ M $ and $ N $ are finitary matroids on the same edge set $ E $, then they admit a common
 independent set $ I $ for which there is a partition $ E=E_M\sqcup E_N $ such that $ I_M:=I\cap E_M $ spans $ E_M $ in $
 M $ and  $ I_N:=I\cap E_N $ spans $ E_N $ in $ N $.
\end{conj}

 Aharoni proved in \cite{aharoni1984konig} based on his earlier works with  Nash-Williams and  Shelah (see
\cite{aharoni1983general} and \cite{aharoni1984another})
that the special case of Conjecture \ref{MIC} where $ M $ and $ N $ are partition matroids holds. The conjecture
is also known
to be true  if we assume that $ E $ is countable but $ M $ and
$ N $ can be otherwise arbitrary (see \cite{joo2020MIC}).
Let us call Generalized Matroid Intersection Conjecture what we obtain from \ref{MIC} by extending it to arbitrary matroids
(i.e. omitting the word ``finitary''). Several partial results has been obtained for this generalization but only for well-behaved
matroid classes. The positive answer is known for example if: $ M $ is finitary and $ N $ is cofinatory
\cite{aigner2018intersection}  or both
matroids are
singular\footnote{A matroid is singular if it is the direct sum of $ 1 $-uniform matroids and duals of $ 1
$-uniform matroids.} and countable \cite{ghaderi2017}  or $ M $ is arbitrary and $ N $ is the direct sum of
finitely many uniform matroids \cite{joó2020intersection}.

Bowler and Carmesin showed that their Pacing/Covering Conjecture \ref{conj: Pack/Cov} and the
Generalized Matroid Intersection Conjecture are equivalent and they also found  an important reduction for both (see Corollary
3.9 in \cite{bowler2015matroid}). By analysing their proof it is clear that the equivalence can be established if we restrict both
conjectures to a class of matroids closed under certain operations. It allows us to prove  Theorem \ref{t: main result} by showing
the following instance of the  Generalized Matroid Intersection Conjecture which itself is a common extension of the singular case
by Ghaderi \cite{ghaderi2017} and our previous work \cite{joo2020MIC}:

\begin{restatable}{thm}{MI}\label{t: main result}
  If $ M $ and $ N $ are matroids in $  \mathfrak{F}\oplus \mathfrak{F}^{*} $ on the same countable edge set $ E $, then they
  admit a common independent set $ I $ for which there is a partition $ E=E_M\sqcup E_N $ such that $ I_M:=I\cap E_M $ spans
  $ E_M $ in $ M $ and  $ I_N:=I\cap E_N $ spans $ E_N $ in $ N $.
\end{restatable}

The paper is organized as follows. In the following section we introduce some notation and fundamental facts about matroids that
are mostly well-know for finite ones. In Section \ref{s: premil} we collect some previous results and relatively easy technical
lemmas in order be able the discuss  later the proof of the main results  without any distraction. Then  in Section
\ref{s: reduction} we reduce the main results to a key-lemma. After these preparations the actual proof begins with
Section \ref{s: augP} by developing and analysing an `augmenting path' type of technique.  Our main principle from this point is
to
handle the
finitary and the
cofinitary parts of matroid $ N $ differently in order to exploit the advantage
of the finiteness of the  circuits and cocircuits respectively.  Equipped with these ``mixed'' augmenting paths we discuss the proof
of our key-lemma in   Section
\ref{s: proof of key-lemma}. Finally, we introduce an application in Section \ref{s: application} about orientations of a graph with
in-degree requirements.

\section{Notation and basic facts}\label{sec notation}

In this section we introduce the notation and recall some basic facts about
matroids that  we will use later without further explanation. For more details we refer to  \cite{nathanhabil}.

An enumeration of a countable set $ X $ is an $ \mathbb{N}\rightarrow X $ surjection that we write as $ \{x_n: n\in \mathbb{N}
\} $.  We denote the symmetric difference $ (X\setminus Y)\cup (Y\setminus X) $ of $ X $ and $ Y $ by $
\boldsymbol{X\vartriangle Y} $.
A pair ${M=(E,\mathcal{I})}$ is a \emph{matroid} if ${\mathcal{I} \subseteq \mathcal{P}(E)}$ satisfies  the axioms
\ref{item axiom1}-\ref{item axiom4}.
The sets in~$\mathcal{I}$ are called \emph{independent} while the sets in ${\mathcal{P}(E) \setminus \mathcal{I}}$ are
\emph{dependent}. An $ e\in E $ is a \emph{loop} if $ \{ e \} $ is dependent.
If~$E$ is finite, then  \ref{item axiom1}-\ref{item axiom3}  is one of the the usual axiomization of matroids in terms of
independent sets  (while \ref{item axiom4} is redundant).
The maximal independent sets are called \emph{bases}. If $ M $ admits a finite base, then all the bases have the same size which
is the rank $ \boldsymbol{r(M)} $ of $ M $ otherwise we let $ r(M):=\infty $.\footnote{It is independent of ZFC that the bases of
a fixed matroid  have the same
size (see \cite{higgs1969equicardinality} and \cite{bowler2016self}).} The minimal dependent sets are called \emph{circuits}.
Every dependent set contains a circuit. The  \emph{components} of a matroid are the
components of the hypergraph of its circuits.  The
\emph{dual} of a matroid~${M}$ is the
matroid~${M^*}$ with $ E(M^*)=E(M) $ whose bases are the complements of
the bases of~$M$. For an  ${X \subseteq E}$, ${\boldsymbol{M  \upharpoonright X} :=(X,\mathcal{I}
\cap \mathcal{P}(X))}$ is a matroid and it  is called the \emph{restriction} of~$M$ to~$X$.
We write ${\boldsymbol{M - X}}$ for $ M  \upharpoonright (E\setminus X) $  and call it the minor obtained by the
\emph{deletion} of~$X$.
The \emph{contraction} of $ X $ in $ M $ and the contraction of $ M $ onto $ X $ are
${\boldsymbol{M/X}:=(M^* - X)^*}$ and $\boldsymbol{M.X}:= M/(E\setminus X) $ respectively.
Contraction and deletion commute, i.e., for
disjoint
$ X,Y\subseteq E $, we have $ (M/X)-Y=(M-Y)/X $.  Matroids of this form are the  \emph{minors} of~$M$. The
independence of an $ I\subseteq X $ in $ M.X $ is equivalent with $ I\subseteq \mathsf{span}_{M^{*}}(X\setminus I) $.
If $ I $ is independent in $ M $  but $ I+e $ is dependent for some $ e\in E\setminus I $  then there is a unique
circuit   $ \boldsymbol{C_M(e,I)} $ of $ M $ through $ e $ contained in $ I+e $.  We say~${X
\subseteq E}$ \emph{spans}~${e \in E}$ in matroid~$M$ if either~${e \in X}$ or there exists a circuit~${C
\ni e}$ with~${C-e \subseteq X}$.
By letting $\boldsymbol{\mathsf{span}_{M}(X)}$ be the set of edges spanned by~$X$ in~$M$, we obtain a closure
operation
$ \mathsf{span}_{M}: \mathcal{P}(E)\rightarrow \mathcal{P}(E) $.
An ${S \subseteq E}$ is \emph{spanning} in~$M$ if~${\mathsf{span}_{M}(S) = E}$. An $ S\subseteq X $ spans $ X $ in $
M.X $ iff $ X\setminus S $ is independent in $ M^{*} $. If $ M_i=(E_i,
\mathcal{I}_i)$ is a matroid for $ i\in
\Theta $ and the sets $ E_i $ are pairwise disjoint, then their direct sum is $ \boldsymbol{\bigoplus_{i\in
\Theta}M_i}=(E,\mathcal{I}) $ where
$ E=\bigsqcup_{i\in \Theta}E_i $ and $ \mathcal{I}=\{ \bigsqcup_{i\in \Theta}I_i : I_i\in \mathcal{I}_i\} $. For a class $
\mathfrak{C} $ of matroids $ \boldsymbol{\mathfrak{C}(E)} $ denotes the subclass $ \{ M\in \mathfrak{C}: E(M)=E \} $.
A matroid is called uniform if for ever base $ B $ and every  edges $ e\in B $ and $ f\in E\setminus B $ the set $ B-e+f $ is also a
base. Let
$ \boldsymbol{U_{E,n}}$ be the $ n $-uniform matroid on $ E $, formally $ U_{E,n}:=(E , [E]^{\leq n})$.

We need some further more subject-specific definitions. From now on let $ M
$ and $ N $ be matroids on a common edge set $ E $.  We call a $ W\subseteq E $ an
$ (M,N) $-\emph{wave} if  $ M\upharpoonright W $ admits an $ N.W $-independent base. Waves in the matroidal context were
introduced by
Aharoni and Ziv in \cite{aharoni1998intersection} but it was also an important tool in the proof of the infinite version of Menger's
theorem \cite{aharoni2009menger} by Aharoni and Berger. We write $ \boldsymbol{\mathsf{cond}(M,N)} $ for
the condition: `For every $ (M,N) $-wave $ W $ there is an $ M $-independent base of $ N.W $.' A set $ I\in\mathcal{I}_M\cap
\mathcal{I}_N $ is \emph{feasible} if $
\mathsf{cond}(M/I,N/I) $ holds.
It is known (see Proposition \ref{wave union})  that there exists a $ \subseteq $-largest $ (M,N) $-wave which we denote  by
$ \boldsymbol{W(M,N)} $.  Let $ \boldsymbol{\mathsf{cond}^{+}(M,N) }$ be the statement that $ W(M,N) $ consists of $ M
$-loops and $
{r(N.W(M,N))=0} $. As the notation indicates it is a strengthening of $ \mathsf{cond}(M,N) $. Indeed, under  the assumption $
\mathsf{cond}^{+}(M,N) $,  $ \varnothing $ is an $ M $-independent base of $ N.W $ for every wave $ W $. A feasible $ I $ is
called  \emph{nice} if $ \mathsf{cond}^{+}(M/I,N/I)  $ holds. For $ X\subseteq E $ let $ \boldsymbol{B(M,N,X)} $ be the
(possibly empty) set of common bases of $ M \upharpoonright
X $ and $ N.X $.
\section{Preliminary lemmas and preparation}\label{s: premil}
We collect  those necessary lemmas in this section that are either known from previous papers or follow more or less directly
from definitions.

\subsection{Classical results}
The following two statements were proved by Edmonds' in \cite{edmonds2003submodular}:
\begin{prop}\label{prop: simult change}
        Assume that~$I$ is independent,
        ${e_1, \dots, e_{m} \in \mathsf{span}(I) \setminus I}$
        and~${f_1, \dots, f_{m} \in I}$
        with ${f_j \in C(e_j, I)}$
        but ${f_j\notin C(e_k, I)}$ for~${k < j}$.
        Then
        \[
            {\left( I \cup \{e_1,\dots, e_{m}\} \right) \setminus \{f_1,\dots, f_{m}\}}
        \]
        is independent and spans the same set as~$I$.
    \end{prop}
    \begin{proof}
        We use induction on~$m$.
        The case~${m = 0}$ is trivial.
        Suppose that~${m > 0}$.
        On the one hand, the set ${I-f_m+e_m}$ is independent and spans the same set as~$I$.
        On the other hand, ${C(e_j, I-f_m+e_m) = C(e_j, I)}$ for~${j < m}$ because ${f_m \notin C(e_j, I)}$ for~${j < m}$.
        Hence by using the induction hypothesis for~${I-f_m+e_m}$ and ${e_1,\dots, e_{m-1}, f_1,\dots, f_{m-1}}$ we are done.
    \end{proof}
\begin{lem}[Edmonds' augmenting path method]\label{l: augP Edmonds}
For $ I\in \mathcal{I}_M\cap \mathcal{I}_N $, exactly one of the following statements holds:
\begin{enumerate}
\item There is a partition $ E=E_M\sqcup E_N $ such that $ I_M:=I\cap E_M $ spans $ E_M $ in $ M $ and  $ I_N:=I\cap E_N $
spans $ E_N $ in $ N $.

\item There is a
 $ P=\{ x_1,\dots, x_{2n+1} \}\subseteq E $  with $ x_{1}\notin \mathsf{span}_N(I) $ and $
 x_{2n+1}\notin
 \mathsf{span}_M(I) $ such that $ I\vartriangle P\in
 \mathcal{I}_M\cap \mathcal{I}_N $ with $ \mathsf{span}_{M}(I\vartriangle P) =\mathsf{span}_{M}(I+x_{2n+1}) $ and
 $ \mathsf{span}_{N}(I\vartriangle P) =\mathsf{span}_{M}(I+x_{1}) $.
\end{enumerate}
\end{lem}
\noindent We will develop in Section
\ref{s: augP}  a ``mixed'' augmenting path method which operates differently on the finitary
and on the cofinitary part of an $ N\in (\mathfrak{F}\oplus \mathfrak{F}^{*})(E) $. The phrase `augmenting path' refers always
to our mixed method except in the proof of Lemma \ref{one more edge}.
Note that $ E_M $ is an $ (M,N) $-wave witnessed by $ I_M $ and $ E_N $ is an $ (N,M) $-wave witnessed by $
I_N $.

 One can define matroids in the language of circuits (see \cite{bruhn2013axioms}). The following claim is one of the axioms in
 that case.
\begin{claim}[Circuit elimination axiom]\label{Circuit elim}
Assume that $ C\ni e $ is a circuit and $ \{ C_x: x\in X \} $ is a family of circuits where $ X\subseteq C-e $ and $ C_x $ is a
circuit with $C\cap X=\{ x \} $ avoiding $ e $. Then there is a circuit through $ e $ contained in
 \[ \left( C\cup \bigcup_{x\in X}C_x \right) \setminus X =:Y \]
\end{claim}
\begin{proof}
Since $ C_x-x$ spans $ x $ we have $ C-e\subseteq\mathsf{span}(Y-e) $ and therefore $e\in \mathsf{span}(\mathsf{span}(Y-e)
)$. But
then $ e\in \mathsf{span}(Y-e) $ because $ \mathsf{span} $ is a closure operator.
\end{proof}
For finite matroids the axiom above is demanded only in the special case  where $ X $ is a singleton (known as ``Strong circuit
elimination``) from which the case of arbitrary $ X $ can be derived by repeated application.
\begin{cor}\label{cor: Noutgoing arc}
Let $ I $ be an independent and suppose that there is a circuit $ C\subseteq \mathsf{span}(I) $
with $ e\in I\cap C $.  Then there is an $ f\in C\setminus I $ with $e\in  C(f,I) $.
\end{cor}
\begin{proof}
For every $ x\in C\setminus I $ we pick a circuit $ C_x $ with $ C_x\setminus I=\{ x \} $. If $
e\in C_x $
for some $ x $,
then $ f:=x $ is as desired. Suppose for a contradiction that there is no such an $ x $. Then by Circuit elimination (Claim
\ref{Circuit elim}) we obtain a circuit through $ e $ which is
contained entirely in $ I $ contradicting the independence of $ I $.
\end{proof}

The following statement was shown by Aharoni and Ziv in \cite{aharoni1998intersection} using a slightly
different terminology.
\begin{prop}\label{wave union}
The union of arbitrary many waves is a wave.
\end{prop}
\begin{proof}
Suppose that $ W_{\beta} $ is a wave  for $ \beta<\kappa $ and let
$W_{<\alpha}:=\bigcup_{\beta<\alpha}W_{\beta}  $ for $ \alpha\leq \kappa $. We  fix a base
$ B_{\beta} \subseteq W_{\beta} $
of $ M\upharpoonright W_{\beta} $ which is independent in $ N.W_{\beta} $.  Let us define
$ B_{<\alpha} $  by transfinite recursion for $ \alpha\leq \kappa $ as follows.

\[B_{<\alpha}:= \begin{cases} \varnothing &\mbox{if } \alpha=0 \\
B_{<\beta}\cup (B_\beta \setminus W_{<\beta}) & \mbox{if } \alpha=\beta+1\\
\bigcup_{\beta<\alpha}B_{<\beta} & \mbox{if } \alpha \text{ is limit ordinal}.
\end{cases} \]

 First we show  by transfinite induction
 that $ B_{<\alpha} $ is spanning in $ M\upharpoonright W_{<\alpha} $. For $ \alpha=0 $ it is trivial.
 For a limit $ \alpha $ it follows directly from the induction hypothesis. If $ \alpha=\beta+1 $, then
 by the choice of $ B_\beta $, the set $ B_\beta \setminus W_{<\beta} $ spans $ W_{\beta}\setminus W_{<\beta} $
 in $ M/W_{<\beta} $. Since $ W_{<\beta} $ is spanned by $ B_{<\beta} $ in $ M $ by induction, it follows that
 $ W_{<\beta+1} $ is spanned by $ B_{<\beta+1} $ in $ M $.

 The  independence of $ B_{<\alpha} $ in
 $ N.W_{<\alpha} $   can be reformulated as ``$W_{<\alpha}\setminus B_{<\alpha}$ is spanning in $
 N^{*} \upharpoonright
 W_{<\alpha} $'',
 which can be proved the same way as above.
\end{proof}

\subsection{Some more recent results and basic facts}

\begin{thm}[Aigner-Horev, Carmesin and Frölich; Theorem 1.5 in \cite{aigner2018intersection}]\label{t: mixed}
If $ M\in \mathfrak{F}(E) $ and $ N\in
\mathfrak{F}^{*}(E) $, then there is an $ I\in \mathcal{I}_M\cap \mathcal{I}_N $ and a partition $ E=E_M\sqcup E_N $ such
that $ I_M:=I\cap E_M $ spans $ E_M $ in $ M $ and  $ I_N:=I\cap E_N $ spans $ E_N $ in $ N $.
\end{thm}

\begin{cor}\label{cor: applyMixed}
If $ M\in \mathfrak{F}(E) $ and $ N\in
\mathfrak{F}^{*}(E) $ satisfy $ \mathsf{cond}^{+}(M,N) $, then there is an $ M $-independent $ N $-base.
\end{cor}
\begin{proof}
Let $ E_M, E_N,
I_M $ and $ I_N $ be as in Theorem  \ref{t: mixed}. Then $ E_M $ is a wave witnessed by $ I_M $ thus by
$ \mathsf{cond}^{+}(M,N) $ we know that $ E_M $ consists of $ M $-loops and $ r(N.E_M)=0 $. But then $ I_M=\varnothing $
and $ I_N $ is a base of $ N $ which is independent in $ M $.
\end{proof}

\begin{obs}\label{o: Mloop}
If $ \mathsf{cond}(M,N) $ holds and $ L $ is a set of the $ M $-loops, then $ r(N.L)=0 $ which means $
L\subseteq\mathsf{span}_N(E\setminus L) $.
\end{obs}
\begin{cor}\label{cor: Mloop}
If $ I $ is feasible, then $ r(N.(\mathsf{span}_{M}(I)\setminus I)=0$.
\end{cor}

\begin{obs}
If $ W_0 $ is an $ (M,N) $-wave and $ W_1 $ is an $ (M/W_0, N-W_0) $-wave, then $ W_0\cup W_1 $ is an $ (M,N)
$-wave.
\end{obs}

\begin{cor}\label{cor: empty wave}
For $ W:=W(M,N) $, the largest $ (M/W, N-W) $-wave  is $ \varnothing $. In particular, $ \mathsf{cond}^{+}(M/W,N-W) $
holds.
\end{cor}

\begin{obs}\label{o: condRestrict}
 $ \mathsf{cond}^{+}(M,N) $ implies ${\mathsf{cond}^{+}(M\upharpoonright X,N.X) }$  for every $ X\subseteq E $.
\end{obs}
\begin{prop}\label{p: iterate feasible}
If $ I_0\in \mathcal{I}_N\cap \mathcal{I}_M $ and $ I_1 $ is feasible with respect to $ (M/I_0,
N/I_0) $, then $ I_0\cup I_1 $ is
feasible with respect to $ (M,N) $. If in addition $ I_1 $ is a nice feasible in regards to $ (M/I_0, N/I_0) $, then so is
$ I_0\cup I_1 $ to $ (M,N) $.
\end{prop}
\begin{proof}
By definition the feasibility of $ I_1 $ w.r.t. $ (M/I_0, N/I_0) $ means  that the condition $ \mathsf{cond}(M/(I_0\cup
I_1),N/(I_0\cup I_1)) $
holds. The feasibility of $ I_0\cup I_1 $ w.r.t. $ (M, N) $ means the same also by definition. For `nice feasible'  the argument is
similar, only  $ \mathsf{cond} $ must be replaced by $ \mathsf{cond}^{+} $.
\end{proof}

The following lemma was introduced in  \cite{joo2020MIC}.
\begin{lem}\label{one more edge}
Condition $ \mathsf{cond}^{+}(M, N) $ implies that
 whenever $ W $ is an $ (M/e, N/e) $-wave for some $ e\in E $ witnessed by $ B\subseteq W $, then
 $B\in  B(M/e,N/e,W) $, i.e. $ B $ is spanning in $ N.W $.
\end{lem}
\begin{proof}
Let $ W $ be an $ (M/e, N/e)  $-wave. Note that
$(N/e).W=N.W  $ by definition. Pick a $ B\subseteq W $ which is an $ N.W $-independent base of $ (M/e)\upharpoonright
W $.  We may assume
that
$ e\in \mathsf{span}_M(W) $ and $ e $ is not an $ M $-loop. Indeed,  otherwise $  (M/e) \upharpoonright
W=M\upharpoonright
W$ holds
and hence $ W $ is also an $ (M,N) $-wave. Thus by $ \mathsf{cond}^{+}(M, N) $ we may conclude that $ W $ consists of $ M
$-loops  and hence $ B=\varnothing $, moreover, $ r(N.W)=0 $ and therefore $
\varnothing $ is a base of $ N.W $.

Then $ B $ is not a base of
$ M\upharpoonright W $  but ``almost'', namely $r(M/B \upharpoonright W) =1 $.  We apply the
augmenting path Lemma \ref{l: augP Edmonds} by Edmonds  with $ B$ in regards to  $M\upharpoonright W$ and $  N.W  $. An
augmenting
path $ P $ cannot exist. Indeed, if $ P $ were an augmenting path then $ B\vartriangle P $ would show that $ W $ is an
 $ (M,N) $-wave which does not consist of $ M $-loops, contradiction. Thus we get a partition
$ W=W_{0}\sqcup W_{1} $ instead where $ W_0 $ is an $ (M \upharpoonright W,N.W) $-wave witnessed by $ B\cap W_0 $,
and therefore also an $ (M,N)
$-wave,  and $ W_1 $ is an
$ (N.W, M\upharpoonright W) $-wave showed by $ B\cap W_1 $.
Then $ W_0 $ must consist of $ M $-loops by $ \mathsf{cond}^{+}(M, N) $ and
therefore $ B\subseteq W_1 $ by the $ M $-independence of $ B $. We need to show that $ B $ is spanning not just in $ N.W_1 $
but also in $ N.W $. To do so let $ B' $ be a base of $ N-W$.
Then $ B\cup B' $ spans $ E\setminus W_0 $ in $ N $ because $ B=B\cap W_1 $ is a base $ N.W_1 $. But
then $ B\cup B' $ is spanning in  $ N $ because $ r(N.W_0)=0 $ by Observation \ref{o: Mloop}. We conclude that $ B $ is
spanning in $ N.W $ as desired.
 \end{proof}

\subsection{Technical lemmas in regards to \texorpdfstring{$ \boldsymbol{B(M,N,W)} $}{BMNW}}

\begin{prop}\label{prop: WB nice feasible}
For $ W:=W(M,N) $, the elements of $ B(M,N,W) $ are nice feasible sets.
\end{prop}
\begin{proof}
Let $ B\in  B(M,N,W) $. Clearly $ B\in \mathcal{I}_M\cap \mathcal{I}_N $ because $ B\in \mathcal{I}_M\cap
\mathcal{I}_{N.W} $ by definition and $ \mathcal{I}_{N.W} \subseteq \mathcal{I}_N $.  We know that $ W\setminus B $ is
an $
(M/B, N/B) $-wave consisting of $ M/B $ loops and $ r(N.(W\setminus B))=0 $ because $ B $ is a
base of $ N.W $. In order to show $ \mathsf{cond}^{+}(M/B, N/B)
$, it is enough to prove that $ W\setminus B $ is actually  $ W(M/B, N/B)=:W' $. Suppose for a contradiction that $ W'\supsetneq
W\setminus B$. Let $ B' $ be an $ N.W' $-independent base of $ M/B \upharpoonright W' $. Note that $ B\cup B' $ is a base of $
M \upharpoonright (W\cup
W') $ and $ B'\cap (W\setminus B)=\varnothing $. Since $ B $ is $ N^{*} $-spanned by $ W\setminus B\subseteq (W\cup
W')\setminus (B\cup B') $ and $ B' $ is $ N^{*} $-spanned by $ W'\setminus B'\subseteq (W\cup W')\setminus (B\cup B')  $, we
may conclude that $ B\cup B' $ is independent in  $ N.(W\cup W') $. Thus $ W\cup W' $ is an $ (M,N) $-wave witnessed by $
B\cup B' $ which contradicts the maximality of $ W $.
\end{proof}
\begin{cor}\label{cor: extNice}
Assume that $ I\in \mathcal{I}_M\cap \mathcal{I}_M $ and $ B\in B(M/I,N/I, W) $ where  $ W:=W(M/I,N/I) $.  Then $ I\cup B
$ is a nice feasible set.
\end{cor}
\begin{proof}
Combine Propositions \ref{p: iterate feasible} and \ref{prop: WB nice feasible}.
\end{proof}

\begin{obs}\label{obs: cond+ loop delete}
If $ \mathsf{cond}^{+}(M,N) $ holds and $ L $ is a set of $ M $-loops, then $ \mathsf{cond}^{+}(M-L,N-L) $ also holds.
\end{obs}

 \begin{obs}\label{obs: common loops remove}
 Let $ W $ be a wave and let $ L $ be a set of common loops of $ M $ and $ N $. Then $ W\setminus L $ is also a wave and $
 B(M,N,W)=B(M,N,W\setminus L) $.
 \end{obs}
\begin{lem}\label{l: wave modify}
Let $ W $ be a wave and $  L\subseteq W $ such that $ L $ consists of $ M $-loops with $ r(N.L)=0 $. Then $ W\setminus L $ is
an $ (M-L, N-L, W\setminus L) $-wave with \[
B(M,N,W)=B(M-L, N-L, W\setminus L). \]
\end{lem}
\begin{proof}
A set $ B $ is a base of  $ M\upharpoonright W $ iff it is a base of
$ M\upharpoonright (W\setminus L) $ because $ L $ consists of $ M $-loops. A $ B\subseteq W\setminus L $ is $ N.W
$-independent iff $ B\subseteq\mathsf{span}_{N^{*}}(W\setminus B) $.  This holds if and only if  $
B\subseteq\mathsf{span}_{N^{*}/L}(W\setminus
(B\cup L)) $ i.e. $ B  $ is  independent in $ (N-L).(W\setminus L) $.   Note that $ r(N.L)=0 $ is equivalent with the $
N^{*} $-independence of $ L $. Thus for $ B\subseteq W\setminus L $,  $ W\setminus B $ is $ N^{*} $-independent iff
$ W\setminus (B\cup L) $ is $ N^{*}/L $-independent. It means that $ B $ is spanning in $ N.W $ iff it is spanning in $
(N-L).(W\setminus L) $.

Thus the sets that are witnessing that $ W\setminus L $ is
an $ (M-L, N-L, W\setminus L) $-wave are exactly those that are witnessing  that  $ W $ is an $ (M,N) $-wave, moreover, $
B(M,N,W)=B(M-L, N-L, W\setminus L) $ holds.

\end{proof}

\begin{lem}\label{l: minorsChanged}
Assume that $ X_j, Y_j \subseteq E $ for $ j\in \{ 0,1 \} $ where $ X_j\sqcup Y_j=Z $ for $ j\in \{ 0,1 \} $, furthermore
$ \mathsf{span}_M(X_0)=\mathsf{span}_M(X_1) $ and $ \mathsf{span}_{N^{*}}(Y_0) =\mathsf{span}_{N^{*}}(Y_1) $.
Then for every $ X\subseteq  E\setminus Z $ we have\[ B(M/X_0-Y_0, N/X_0-Y_0, X)=B(M/X_1-Y_1, N/X_1-Y_1, X). \]
\end{lem}
\begin{proof}
The matroids $ M/X_0-Y_0 $ and  $ M/X_1-Y_1$ are the same as well as the matroids $ N/X_0-Y_0$ and  $ N/X_1-Y_1 $.
\end{proof}

\section{Reductions}\label{s: reduction}
We repeat here our main results for convenience:
\PC*

\MI*

\subsection{Matroid Intersection with a finitary \texorpdfstring{$\boldsymbol{M} $}{M}}
As we mentioned, the  method by Bowler and Carmesin used to prove  Corollary 3.9 in \cite{bowler2015matroid} works not only
for the class of all matroids but can be adapted for every class  closed under certain operations. We apply their
technique to obtain the following reduction:

\begin{lem}\label{l: M finitary}
Theorems \ref{t: main result0} and \ref{t: main result} are implied by the special case of Theorem \ref{t: main result} where $
M\in \mathfrak{F} $.
\end{lem}

\begin{proof}
First we show that one can assume without loss of generality in the proof of Theorem \ref{t: main result0} that $ \Theta  $ is
countable. To
do so let  \[ E':=\{ e\in E: \left|\{i\in
\Theta: \{ e \}\in \mathcal{I}_{M_i} \}\right|\leq \aleph_0 \} \]  and
\[  \Theta':= \{ i\in \Theta: (\exists e\in E' ) (\{ e \}\in \mathcal{I}_{M_i}) \}. \]
We apply Theorem \ref{t: main result0} with $ E' $ and with the  countable family  $(M_i\upharpoonright E': i\in \Theta') $.
Then we obtain a partition
$ E'=E'_p\sqcup E'_c $ such that $ (M_i \upharpoonright E'_p: i\in \Theta') $ admits a packing $ (S_i: i\in \Theta') $ and
$ (M_i\upharpoonright E'. E_c: i\in \Theta') $ admits a covering $ (I_i: i\in \Theta') $. Let $ E_p:=E'_p $ and $ E_c:= E\setminus
E_p=E'_c\cup (E\setminus E')  $. By construction $ r_{M_i}(E')=0 $ for $ i\in
\Theta \setminus \Theta' $. Thus by letting $ S_i:=\varnothing $  for $ i\in \Theta \setminus \Theta' $ the
family $ (S_i: i\in \Theta) $ is a packing w.r.t. $ (M_i \upharpoonright E_p: i\in \Theta) $.
Let  $ g: E\setminus E'\rightarrow
\Theta\setminus \Theta' $ be injective. For $ i\in \Theta \setminus \Theta'  $, we take $ I_i:=\{ g^{-1}(i) \}$ if $ i\in
\mathsf{ran}(g) $ and $ I_i:=\varnothing $ otherwise. Then $ (I_i: i\in \Theta) $ is a covering for $ (M_i. E_c: i\in \Theta) $ and
we are done.

We proceed with the proof of Theorem \ref{t: main result0} assuming  that $ \Theta $ is countable.  For $ i\in \Theta $, let $ M'_i
$ be the matroid on $
E\times \{ i \} $ that we
obtain by
``copying'' $ M_i $ via the bijection $ e\mapsto (e,i) $.  Then for

\[M:= \bigoplus_{i\in \Theta}M_i',\ \text{ and } N:= \bigoplus_{e\in E}U_{ \{ e \}\times \Theta, 1  } \]
we have $M\in ( \mathfrak{F}\oplus \mathfrak{F}^{*})(E\times \Theta) $ and $ N\in \mathfrak{F}(E\times \Theta) $ where $
E\times \Theta $ is countable. Thus by assumption there is a partition $
E\times \Theta=E_M\sqcup E_N $ and an $ I\in \mathcal{I}_{M}\cap \mathcal{I}_{N} $ such that $ I_M:=I\cap E_M $
spans $ E_M $ in $ M$ and $ I_N:=I\cap E_N $
spans $ E_N$ in $ N$. The  $ M $-independence of $ I $ ensures that $ J_i:=\{ e\in E: (e,i)\in I \} $ is $ M_i $-independent.
The $ N $-independence of $ I $ guarantees  that the sets $ J_i $ are pairwise disjoint. Let
$ E_c:=\{ e\in E: (\exists i\in \Theta) (e,i)\in E_N \} $. Then for each $ e\in E_c $ there must be some $ i\in \Theta $ with $
(e,i)\in
I_N $ because $ E_N\subseteq \mathsf{span}_N(I_N) $. Thus the sets $ J_i $ cover $ E_c $ and so do the sets $
I_i:=J_i\cap E_c $. It is enough to show that $ S_i:=J_i\setminus I_i $ spans $ E_p:=E\setminus E_c $ in $ M_i $ for every $
i\in \Theta $. Let $ f\in E_p $ and $ i\in \Theta $ be given.  Then $ \{ f \}\times \Theta \subseteq E_M $ follows directly from the
 definition of $ E_p $, in particular  $ (f,i)\in E_M $. We know that $(f,i)\in
  \mathsf{span}_{M}(I_M)$ and hence  $ f\in \mathsf{span}_{M_i}(\{ e\in E: (e,i)\in I_M \}) $. Suppose for a contradiction that
  $h\in E_c\cap \{ e\in E: (e,i)\in I_M \} $. Then for some $ j\in
  \Theta $ we have $ (h,j)\in E_N $. Since $ (h,j)\in \mathsf{span}_N(I_N) $, we have $ (h,k)\in I_N $ for some $ k\in \Theta $.
  But then $ (h,i), (h,k)\in I $ are distinct elements thus $ i\neq k $ which contradict the $ N $-independence of $ I $. Therefore $
  E_c\cap \{ e\in E: (e,i)\in I_M \}=\varnothing $. Since $ \{ e\in E: (e,i)\in I_M  \}\subseteq J_i $  by the definition of $ J_i
  $ we conclude $  \{ e\in E: (e,i)\in I_M \}=S_i $. Therefore $ (S_i: i\in \Theta) $ is a packing for $ (M_i \upharpoonright E_p:
  i\in \Theta) $ and $ (I_i: i\in \Theta) $  is a covering for $ (M_i.  E_c: i\in \Theta) $ as desired.

Now we derive Theorem \ref{t: main result} from Theorem \ref{t: main result0}. To do so, we take a partition $E=E_p\sqcup
E_c$ such that $ (S_M, S_N) $ is a packing for
 $ (M\upharpoonright E_p, N^{*}\upharpoonright E_p) $ and $ (R_M, R_N) $ is a covering for $ (M.E_c, N^{*}.E_c ) $.  Let $
 I_M\subseteq S_M $ be a base of  $ M\upharpoonright E_p $ and we define $ I_N:= R_M $. By construction
  $E_p\subseteq \mathsf{span}_M(I_M)  $ and $ I_N\in \mathcal{I}_{M.E_c} $.    We also know that
 \[ I_M\subseteq \mathsf{span}_{N^{*}}(S_N) \subseteq \mathsf{span}_{N^{*}}(E_p\setminus I_M) \]
 which means $ I_M\in \mathcal{I}_{N.E_{p}} $.
 Finally, $R_N\in \mathcal{I}_{N^{*}.E_c} $ means that $ E_c\setminus R_N $ spans $ E_c $ in $ N $ and therefore so does $
 I_N=R_M\supseteq E_c\setminus R_N $.
 \end{proof}
\subsection{Finding an \texorpdfstring{$ \boldsymbol{M} $}{M}-independent base of  \texorpdfstring{$ \boldsymbol{N}
$}{N}}
The following reformulation of the matroid intersection problem was introduced by Aharoni and Ziv in
\cite{aharoni1998intersection} but its analogue by Aharoni was already an important tool to attack (and eventually solve in
\cite{aharoni2009menger}) the Erdős-Menger Conjecture.
\begin{restatable}{claim}{indepB}\label{c: M-indep N-base}
  Assume that  $ M\in \mathfrak{F}(E) $ and $ N\in (\mathfrak{F}\oplus\mathfrak{F}^{*})(E) $ such that $ E $ is countable and
   $ \mathsf{cond}^{+}(M,N) $ holds. Then there is an $ M $-independent base of $ N $.
\end{restatable}
\begin{lem}\label{l: reduc2}
Claim \ref{c: M-indep N-base} implies our main results Theorems \ref{t: main result0} and \ref{t: main result}.
\end{lem}
\begin{proof}
By Lemma \ref{l: M finitary} it is enough to show that the special case of Theorem \ref{t: main result} where $ M\in
\mathfrak{F} $ follows from Claim \ref{c: M-indep N-base}.
To do so, let  $ E_M:=W(M,N) $ and let $ I_M\subseteq E_M $ be a witness that $ E_M $ is a wave. For $ E_N:=E\setminus
E_M $, we have $
M/E_M\in \mathfrak{F}(E_N) $  and
$ N-E_M\in (\mathfrak{F}\oplus \mathfrak{F}^{*})(E_N) $, furthermore, $ \mathsf{cond}^{+}(M/W,N-W) $ holds (see
Corollary \ref{cor: empty wave}). By Claim \ref{c: M-indep N-base}, there is an $ M/E_M $-independent base $ I_N $ of $
N-E_M $. Then $ I\in \mathcal{I}_M\cap \mathcal{I}_{N}$ and $ E=E_M\sqcup E_N $ such that $ I_M:=I\cap E_M $ spans
  $ E_M $ in $ M $ and  $ I_N:=I\cap E_N $ spans $ E_N $ in $ N $ as desired.
\end{proof}
\subsection{Reduction to a key-lemma}
From now on we assume that $ M\in \mathfrak{F}(E) $ and $ N\in (\mathfrak{F}\oplus\mathfrak{F}^{*})(E) $ where $ E $ is
countable.  Let $
\boldsymbol{E_0} $ be
the union of the finitary components of $ N $ and let
 $ \boldsymbol{E_1}:=E\setminus E_0 $.  Note that $ N\upharpoonright E_0 $ is finitary,
$ N\upharpoonright E_1 $ is cofinitary and no $ N $-circuit meets both $ E_0 $ and $ E_1 $.

\begin{restatable}[key-lemma]{lem}{keylemma}\label{l: key-lemma}
  If $ \mathsf{cond}^{+}(M,N) $ holds, then for every $ e\in E_0 $  there is a nice
  feasible $ I $ with $ e\in \mathsf{span}_{N}(I) $.
\end{restatable}

\begin{lem}
Lemma \ref{l: key-lemma}  implies  our main results Theorems \ref{t: main result0} and \ref{t: main result}.
\end{lem}
\begin{proof}
It is enough to show that Lemma \ref{l: key-lemma} implies Claim \ref{c: M-indep N-base} because of  Lemma \ref{l: reduc2}.
Let us
fix an enumeration $ \{ e_n: n\in
\mathbb{N} \} $ of $ E_0 $.
We build an $ \subseteq $-increasing sequence $ (I_n)  $  of  nice feasible sets starting
with $ I_0:=\varnothing $ (who is nice feasible by $ \mathsf{cond}^{+}(M,N) $) in such a way that $ {e_n\in
\mathsf{span}_N(I_{n+1})} $. Suppose that  $ I_n $ is already defined.
If $ e_n\notin \mathsf{span}_N(I_n) $, then we
apply Lemma \ref{l: key-lemma} with $ (M/I_n, N/I_n) $ and $ e_n $ and take the union of the resulting $ I $ with  $ I_{n} $ to
obtain $ I_{n+1} $ (see Observation \ref{p: iterate feasible}), otherwise let $ I_{n+1}:=I_n $. The recursion is done. Now we
construct an $ M $-independent
$ I^{+}_n \supseteq I_n $  with $ E_1\subseteq  \mathsf{span}_N(I^{+}_n) $ for $ n\in \mathbb{N} $. The matroid $ M/I_n
\upharpoonright
(E_1\setminus I_n) $ is finitary and $ N.(E_1\setminus I_n)=(N\upharpoonright E_1)/(I_n\cap E_1)$ is cofinitary, moreover, by
Observation \ref{o: condRestrict}
\[ \mathsf{cond}^{+}(M/I_n, N/I_n) \Longrightarrow \mathsf{cond}^{+}(M/I_n \upharpoonright (E_1\setminus I_n),
N.(E_1\setminus I_n )).  \] Thus by Corollary \ref{cor: applyMixed} there is an $ M/I_n $-independent base $ B_n $ of
$ (N\upharpoonright E_1)/(E_1\cap I_n) $ and $
I^{+}_n:=I_n\cup B_n $ is as desired.

Let $ \mathcal{U} $ be a free ultrafilter on $ \mathcal{P}(\mathbb{N}) $ ad we define
\[ S:=\{ e\in E: \{ n\in \mathbb{N}: e\in I^{+}_n \}\in \mathcal{U} \}. \] Then $ S $ is $ M
$-independent and $ N $-spanning and therefore we are done. Indeed, suppose for a contradiction that $ S $ contains an $ M
$-circuit $ C $. For $ e\in C $, we pick a $ U_e\in
\mathcal{U} $ with $ e\in I^{+}_n $ for $ n\in U_e $. Since $ M $ is finitary, $ C $ is finite, thus $U:= \cap \{ U_e: e\in C \}\in
\mathcal{U} $.
But then for $ n\in U $ we have $ C\subseteq I^{+}_n $ which contradicts the $ M $-independence of $ I^{+}_n $. Clearly, $
I_{n+1}\subseteq S $ for every $ n\in \mathbb{N} $ and therefore $ E_0\subseteq \mathsf{span}_N(S) $. Finally, suppose for a
contradiction that there is some
$ N^{*}\upharpoonright E_1 $-circuit $ C' $ with $ S\cap C'=\varnothing $. For $ e\in C' $, we can pick a $ U_e'\in
\mathcal{U} $ with $ e\notin I^{+}_n $ for $ n\in U_e' $.   Since $ N^{*}\upharpoonright E_1 $ is finitary, $ C' $ is finite, thus
$U':= \cap \{ U_e': e\in
C' \}\in
\mathcal{U} $.
But then for $ n\in U' $ we have $  I^{+}_n\cap C=\varnothing $ which contradicts   $E_1\subseteq
\mathsf{span}_N(I^{+}_n) $.
\end{proof}
\section{Mixed augmenting paths}\label{s: augP}
In the section we  introduce an `augmenting path' type of method and analyse it in order to show some properties we need later.
On $ E_0 $ the definition will be
the same  as in the  proof of
the Matroid Intersection Theorem by Edmonds \cite{edmonds2003submodular} but on $ E_1 $ we need to define it in a different
way considering that $ N\upharpoonright E_1 $ is cofinitary. For brevity we write
$ \boldsymbol{\overset{\circ}{\mathsf{span}}_{M}(F)} $ for $ \mathsf{span}_{M}(F)\setminus F $
and  $ \boldsymbol{F^{j}} $ for $F\cap E_j $
where $ F\subseteq E $ and $ j\in \{ 0,1 \} $.
We call an $F\subseteq E $ \emph{dually safe} if  $ F^{1} $ is spanned by $
\overset{\circ}{\mathsf{span}}_M(F) $ in $ N^{*} $.

\begin{lem}\label{l: enoughAddB}
If $ I\in \mathcal{I}_M\cap \mathcal{I}_n $ is dually safe and  $ B\in B(M/I, N/I, W) $ for $ W:=W(M/I, N/I) $, then
$ I\cup B $ is a nice dually safe feasible set.
\end{lem}
\begin{proof}
We already know by Corollary \ref{cor: extNice} that $ I\cup B $ is a nice feasible set. By using that $ I $ is dually safe and $
\overset{\circ}{\mathsf{span}}_M(I)\subseteq\overset{\circ}{\mathsf{span}}_M(I\cup B) $ we get
\[ I^{1}\subseteq \mathsf{span}_{N^{*}}(\overset{\circ}{\mathsf{span}}_M(I))\subseteq
\mathsf{span}_{N^{*}}(\overset{\circ}{\mathsf{span}}_M(I\cup B)). \]

Since $ B\in B(M/I, N/I, W) $ we have $ W\setminus B\subseteq \overset{\circ}{\mathsf{span}}_M(I\cup B) $ and $ B $ is a
base of $ N.W $. The latter can be rephrased as `$ W\setminus B $ is a base of $ N^{*}\upharpoonright W $'. By combining
these $ \overset{\circ}{\mathsf{span}}_M(I\cup B) $  spans $ B $ in $ N^{*} $. Therefore
$(I\cup B)\cap E_1 \subseteq \mathsf{span}_{N^{*}}(\overset{\circ}{\mathsf{span}}_M(I\cup B)) $, which means that $ I\cup
B $ is dually safe.
\end{proof}
\begin{prop}\label{prop: IE_1 common base}
For a  dually safe feasible $ I $, $ \overset{\circ}{\mathsf{span}}_{M}(I)^{1}  $ is
a base of $ {N^{*}\upharpoonright \mathsf{span}_{M}(I)^{1} } $.
\end{prop}
\begin{proof}
By the definition   of `dually safe', $ \overset{\circ}{\mathsf{span}}_{M}(I)^{1}  $ spans $ N^{*}\upharpoonright
\mathsf{span}_{M}(I)^{1}  $. Furthermore, $ r(N. \overset{\circ}{\mathsf{span}}_{M}(I))=0 $ by Corollary \ref{cor: Mloop},
which is equivalent with the $ N^{*} $-independence of $
\overset{\circ}{\mathsf{span}}_{M}(I) $.
\end{proof}

For a  dually safe  feasible $ I $,
we define an auxiliary digraph $ D(I) $ on $ E $.
Let $ xy $ be an arc of $ D(I) $ iff
one of the following possibilities occurs:
\begin{enumerate}
\item\label{item: D(I) 1} $ x\in E\setminus I $ and $ I+x $ is $ M $-dependent with $ y\in C_M(x, I)-x $,
\item\label{item: D(I) 2} $x\in I^{0}  $ and $ C_{N}(y,I) $ is well-defined and contains $ x $,
\item\label{item: D(I) 3} $x\in I^{1} $ and
$ y\in C_{N^{*}}(x,\overset{\circ}{\mathsf{span}}_{M}(I)^{1} ) -x $ (see
Proposition \ref{prop: IE_1 common base}).
\end{enumerate}

An augmenting path for  a nice dually safe feasible $ I $ is a $ P=\{ x_1,\dots, x_{2n+1} \} $ where
\begin{enumerate}
[label=(\roman*)]
\item $ x_1\in  E_0\setminus \mathsf{span}_N(I) $,
\item $ x_{2n+1}\in E_0\setminus \mathsf{span}_M(I)  $,
\item $ x_kx_{k+1}\in D(I) $ for $ 1\leq k\leq  2n $,
\item\label{item: no jumping} $  x_kx_{\ell}\notin D(I) $ if $ k+1<\ell  $.
\end{enumerate}

\begin{prop}\label{p: augpath}
If $ I $ is a  dually safe feasible set and $ P=\{ x_1,\dots, x_{2n+1} \} $ is an augmenting path for $ I $, then
$ I\vartriangle P$ is a dually safe element of $ \mathcal{I}_M\cap \mathcal{I}_N $  with
\begin{enumerate}
[label=(\Alph*)]
\item\label{item: A} $\mathsf{span}_{M}(I\vartriangle P)=\mathsf{span}_{M}(I+x_{2n+1}) $,
\item\label{item: B} $\mathsf{span}_{N}(I\vartriangle P)\cap E_0=\mathsf{span}_{N}(I+x_1)\cap E_0 $,
\item\label{item: C} $\mathsf{span}_{N^{*}} (\overset{\circ}{\mathsf{span}}_{M}(I)^{1})=\mathsf{span}_{N^{*}}
(\overset{\circ}{\mathsf{span}}_{M}(I)^{1}\vartriangle P^{1}) $ and $
\overset{\circ}{\mathsf{span}}_{M}(I)^{1}\vartriangle P^{1}\in \mathcal{I}_{N^{*}} $.
\end{enumerate}
\end{prop}
\begin{proof}

The set $ I+x_{2n+1} $ is $ M $-independent by the definition of $ P $. Property \ref{item: no jumping} ensures that we can
apply Proposition \ref{prop: simult change} with $I+x_{2n+1},\  e_j=x_{2j-1},\ f_j=x_{2j}\ (1\leq j\leq n) $ and $ M $ and
conclude that
$ I\vartriangle P\in \mathcal{I}_M $ and \ref{item: A} holds.

To prove $ (I\vartriangle P)\cap E_0\in \mathcal{I}_N $ and
\ref{item: B} we
proceed similarly. We start with the $ N $-independent set $ (I+x_{1})\cap E_0 $.  In order to satisfy the premisses of
Proposition \ref{prop: simult change} via property
\ref{item: no jumping}, we need to enumerate the relevant edge
pairs backwards.
Namely, for $j\leq \left| I^{0}\cap P \right|$ let $ e_j:=x_{i_j+1} $ where  $ i_j $ is the $ j $th largest index with $ x_{i_j}\in
I^{0} $ and $ f_j:=x_{i_j} $. We conclude that $ (I\vartriangle P)\cap E_0\in \mathcal{I}_N
$ and \ref{item: B}
holds.

Finally, we let  $ e_j:=x_{i_j} $ for $ j\leq \left|I^{1}\cap P\right| $ where  $ i_j $ is the $j$th smallest  index with $
x_{i_j}\in I^{1} $ and $
f_j:=x_{i_j+1} $.
Recall  that  $ \overset{\circ}{\mathsf{span}}_{M}(I)^{1} $ is $ N^{*} $-independent
(see Proposition  \ref{prop: IE_1 common base}).
We  apply Proposition \ref{prop: simult change} with
$\overset{\circ}{\mathsf{span}}_{M}(I)^{1} ,\  e_j,\ f_j $ and $ N^{*} $ to conclude \ref{item: C}. This means that  $
\overset{\circ}{\mathsf{span}}_{M}(I)^{1}\vartriangle P^{1} $ is a base of $ N^{*}\upharpoonright
\mathsf{span}_{M}(I)^{1}$ because $ \overset{\circ}{\mathsf{span}}_{M}(I)^{1} $ was a base
of it by Proposition \ref{prop: IE_1 common base}.  By \ref{item: A} and by the definition of $ P $ we
know that \[ (I\vartriangle P)\cap
E_1\subseteq
I^{1}\cup P^{1}\subseteq \mathsf{span}^{1}_{M}(I).
\]  By combining these we obtain \[ (I\vartriangle P)\cap
E_1\subseteq \mathsf{span}_{N^{*}}(\overset{\circ}{\mathsf{span}}_{M}(I)^{1}\vartriangle P^{1}). \] The set  $
I\vartriangle
P  $ is disjoint from
$ \overset{\circ}{\mathsf{span}}_{M}(I)\vartriangle P$ because $ I $ is disjoint
from $
\overset{\circ}{\mathsf{span}}_{M}(I) $, moreover, $ \overset{\circ}{\mathsf{span}}_{M}(I)\vartriangle P $ contained in $
\mathsf{span}_{M}(I\vartriangle P) $. Hence  $ \overset{\circ}{\mathsf{span}}_{M}(I\vartriangle P)  $
contains $
\overset{\circ}{\mathsf{span}}_{M}(I)^{1}\vartriangle P^{1} $ and therefore $ N^{*} $-spans
$ (I\vartriangle P) \cap E_1 $, i.e. $ I\vartriangle P $ is dually safe. It means that $ (I\vartriangle P)\cap E_1 $ is independent in $
N.\mathsf{span}_{M}(I\vartriangle P)^{1} $.
Thus uniting   $ (I\vartriangle P)\cap E_0\in \mathcal{I}_N $
with $ (I\vartriangle P)\cap E_1 $ preserves $ N
$-independence, i.e. $ I\vartriangle P\in \mathcal{I}_N $.

\end{proof}
\begin{lem}\label{l: aug extend}
If $ I $ is a nice dually safe feasible set and $ P=\{ x_1,\dots, x_{2n+1} \} $ is an augmenting path for $ I $, then
$ I \vartriangle P $ can be extended to a nice dually safe feasible set.
\end{lem}
\begin{proof}
Let $ M':= M/(I\vartriangle P)$ and $ N':=N/(I\vartriangle P) $.  By Lemma \ref{l: enoughAddB} it is enough to show that
$ B(M', N', W)\neq \varnothing $ for $
W:=W(M', N') $.  Statements \ref{item: A} and \ref{item: B} of Proposition \ref{p: augpath}  ensure that the elements of  $ L:=
\{
x_1, x_3,\dots,
x_{2n-1} \}\cap E_0 $ are common
loops of  $ M'$ and $ N' $.  Statement \ref{item: C} tells that $ Y_0:=\overset{\circ}{\mathsf{span}}_{M}(I)^{1}
$ and $
Y_1:=\overset{\circ}{\mathsf{span}}_{M}(I)^{1}\vartriangle P^{1} $  have the same $ N^{*}
$-span, furthermore, $ Y_1 $ is $ N^{*} $-independent, i.e. $ r(N.Y_1)=0 $. Note  that $ Y_1 $ consists of $ M' $-loops by
\ref{item: A}.
Thus by applying Observation \ref{obs: common loops remove} with $ W $ and $ L $ and then Lemma \ref{l: wave modify}
with  $ W\setminus L $  and $ Y_1 $  we can conclude  that $ W\setminus (L\cup Y_1) $ is an
$ (M'-Y_1, N'-Y_1) $-wave, furthermore,
\[ B(M',N', W)=B(M'-Y_1, N'-Y_1, W\setminus (L\cup Y_1)). \]
The sets $X_0:= I+x_{2n+1} $ and $X_1:= I\vartriangle P $ have the same $ M $-span (see
Proposition \ref{p: augpath}/ \ref{item: A}). Recall
that $ M'=M/X_1 $ and
$
N'=N/X_1 $ by definition. Hence Lemma \ref{l: minorsChanged}
ensures that $ W\setminus (Y_1\cup L) $ is also an $ (M/X_0-Y_0, M/X_0-Y_0) $-wave with
\[ B(M/X_1-Y_1, N/X_1-Y_1, W\setminus (Y_1\cup L))=B(M/X_0-Y_0, M/X_0-Y_0, W\setminus (Y_1\cup L)).  \]
We have $ \mathsf{cond}^{+}(M/I, N/I) $ because $ I $ is a nice feasible set by assumption. Then by
Observation \ref{obs: cond+ loop delete}
$ \mathsf{cond}^{+}(M/I-Y_0, N/I-Y_0) $ also holds. Applying Lemma \ref{one more edge} with $ M/I-Y_0,\  N/I-Y_0,\
W\setminus(Y_0\cup L)   $ and $ x_{2n+1}$ tells
$ B(M/X_0-Y_0, M/X_0-Y_0, W\setminus (Y_1\cup L))\neq \varnothing $ which completes the proof.
\end{proof}

\begin{lem}\label{l: arc remain lemma}
If  $ P=\{ x_1, \dots, x_{2n+1} \} $ is an augmenting path for $ I $ which contains neither  $x$ nor any of its out-neighbours in $
D(I) $, then $xy\in  D(I) $ implies $ xy \in D(I \vartriangle P) $.
\end{lem}
\begin{proof}
Suppose that $ xy\in D(I) $. First we assume that $ x\notin I $. Then the set of the out-neighbours of $ x $ is
 $ C_M(x,I)-x $. By
assumption $ P\cap C_M(x,I)=\varnothing $ and
therefore $C_M(x,I)\subseteq  I \vartriangle P$ thus $ C_M(x,I)=C_M(x,I \vartriangle P) $. This means by definition that $ x $
has the same out-neighbours in
$ D(I) $ and $ D(I \vartriangle P) $.

 If $ x\in I^{1} $, then we can argue similarly.  The set of the out-neighbours of $ x $ in $ D(I) $ is
$ C_{N^{*}}(x,\overset{\circ}{\mathsf{span}}_{M}(I)^{1} ) -x $. By
assumption $ P\cap C_{N^{*}}(x,\overset{\circ}{\mathsf{span}}_{M}(I)^{1} )=\varnothing $ and
therefore  $ C_{N^{*}}(x,\overset{\circ}{\mathsf{span}}_{M}(I)^{1} ) \cap  (I \vartriangle P)=\{ x\} $.  Since $
\mathsf{span}^{1}_{M}(I\vartriangle P) \supseteq
\mathsf{span}^{1}_{M}(I)$ because of Proposition \ref{p: augpath}/\ref{item: A}, we also have $
C_{N^{*}}(x,\overset{\circ}{\mathsf{span}}_{M}(I)^{1} )\subseteq
\mathsf{span}^{1}_{M}(I\vartriangle P) $. By combining these we conclude

 \[ C_{N^{*}}(x,\overset{\circ}{\mathsf{span}}_{M}(I\vartriangle P)^{1} )
 =C_{N^{*}}(x,\overset{\circ}{\mathsf{span}}_{M}(I)^{1} ). \] This means
by definition that $ x $ has the same out-neighbours in $ D(I) $ and $ D(I \vartriangle P) $.

We turn to the case where $ x\in I^{0} $. By definition $ C_N(y,I) $ is well-defined and
contains $ x $,  in particular $ y\in E_0 $.
For $ k\leq n $, let us denote
$ I+x_1-x_2+x_3-\hdots -x_{2k}+x_{2k+1} $ by $ I_k $. Note  that
$ {I_n=I \vartriangle P} $. We show by induction on $ k $ that $ I_k $ is $ N $-independent and
$ {x\in C_N(y, I_k)} $. Since
$ I+x_1 $ is $ N $-independent by definition and $ x_1\neq y $ because $ y\notin P$ by assumption, we obtain $
C_N(y,I)=C_N(y,I_0) $, thus for $ k=0 $ it holds.  Suppose that $ n>0 $ and
we already know the statement for some  $ k<n $. We have $ C_N(x_{2k+3},I_k)=C_N(x_{2k+3},I)\ni x_{2k+2} $ because
there is no ``jumping arc''   in the augmenting path by property \ref{item: no jumping}.  It  follows via the $ N $-independence of
$ I_k $ that  $ I_{k+1} $ is also $ N
$-independent.
If $ x_{2k+2}\notin C_N(y, I_k) $ then $ C_N(y, I_k)=C_N(y, I_{k+1}) $
and the induction step is done. Suppose that   $ x_{2k+2}\in C_N(y, I_k) $. Then $x_{2k+2}, x_{2k+3}\in E_0  $, moreover,
$x\notin C_N(x_{2k+3},I) $ since
otherwise $P$ would contain the out-neighbour $x_{2k+3} $ of $ x $ in $ D(I) $. We apply  circuit elimination (Claim
\ref{Circuit elim})
with  $C= C_N(y, I_k),\ e=x,\  X=\{ x_{2k+2} \},\  C_{x_{2k+2}}=C_N(x_{2k+3},I_k) $. The resulting circuit $ C'\ni x $ can
have at most one element out of $ I_{k+1} $, namely $ y $. Since $ I_{k+1} $ is $ N $-independent, there must be  at least one
such an element and therefore $ C'=C_N(y,I_{k+1}) $.
\end{proof}
\begin{obs}\label{arc remain fact}
If $ xy \in D(I) $ and  $ J\supseteq I $ is a dually safe feasible set with
$ \{ x,y \}\cap J=\{ x, y \}\cap I $, then $ xy \in D(J) $ (the same circuit is the witness).
\end{obs}

\section{Proof of the key-lemma}\label{s: proof of key-lemma}

\keylemma*
\begin{proof}
It is enough to build a  sequence $(I_n)$ of nice dually safe feasible sets  such that
$ (\mathsf{span}_N(I_n)\cap E_0) $ is  an ascending sequence  exhausting $ E_0 $.   We fix a well-order $ \boldsymbol{\prec} $
of type $ \left|E_0\right| $ on  $ E_0 $. Let $ I_0=\varnothing $, which is a nice dually safe feasible set by  $
\mathsf{cond}^{+}(M,N) $.
Suppose that  $ I_n $ is already defined. If there is no augmenting path for $ I_n $, then we let $ I_m:=I_n $ for $ m>n $.
Otherwise we pick an augmenting path $ P_n $ for $ I_n $ in such a way that its first element is as $ \prec $-small as possible.
Then we  apply Lemma \ref{p: augpath} to extend $ I\vartriangle P $  to a nice dually safe feasible set which we define to be $
I_{n+1} $. The recursion is done.

   Let $ \boldsymbol{X}:=E\setminus \bigcup_{n\in \mathbb{N}}\mathsf{span}_{N}(I_n) $ and for $ x\in X $, let $
   \boldsymbol{E(x,n)} $ be the set of elements that are
reachable from $ x $ in $ D(I_n) $ by a directed path. We define $\boldsymbol{n_x} $ to be the smallest
natural number such that  for every $ y\in E\setminus X $ with   $ y \prec x $ we have $ y\in \mathsf{span}_N(I_{n_x}) $.
We shall prove that  \[\boldsymbol{W}:= \bigcup_{x\in X}\bigcup_{n\geq n_x}E(x,n) \] is a wave.

\begin{lem}\label{l: stabilazing stuff}
For every $ x\in X $ and  $  \ell\geq m\geq n_x $,
\begin{enumerate}
\item\label{item stabilize} $ I_m\cap  E(x,m)=I_{\ell}\cap E(x,m) $,
\item\label{item same circuit} $ C_M(y,I_\ell)=C_M(y,I_{m})\subseteq E(x,m)$ for every $y\in  E(x,m)\setminus I_m $,
\item \label{item same cocircuit} $C_{N^{*}}(y,\overset{\circ}{\mathsf{span}}_{M}(I_\ell)^{1})=
C_{N^{*}}(y,\overset{\circ}{\mathsf{span}}_{M}(I_{m})^{1})  $ for every $y\in  E(x,m)\cap I_m^{1} $,
\item\label{item subdigraph} If $ yz\in  D(I_m)$ with $y,z\in E(x,m)$, then  $yz\in  D(I_\ell) $,
\item\label{item increasing}  $ E(x,m)\subseteq E(x,\ell) $.
\end{enumerate}
\end{lem}
\begin{proof}
Suppose that there is an $ n\geq n_x $ such that we know already the statement  whenever  $m,\ell \leq n $. For the induction step
it is
enough to
show that the claim holds for $ n $ and $ n+1 $. We may assume that $ P_n $ exists, i.e. $ I_n\neq I_{n+1} $, since otherwise
there is nothing to prove.

\begin{prop}\label{p: nx no aug}
 $ P_n \cap E(x,n)=\varnothing $.
\end{prop}
\begin{proof}
A common element of  $ P_n $ and $ E(x,n) $ would show that there is also an augmenting path in $ D(I_n) $ starting at $ x $
which is impossible since $ x\in X $ and  $ n\geq n_x $.
\end{proof}
\begin{cor}
$ I_{n}\cap  E(x,n)=(I_n \vartriangle P_n)\cap E(x,n) $.
\end{cor}
\begin{prop}
 $ (I_n \vartriangle P_n)\cap  E(x,n)=I_{n+1}\cap E(x,n) $.
\end{prop}
\begin{proof}
If $y\in  E(x,n)\setminus I_n  $, then its out-neighbours in $ D(I_n) $ are in $ E(x,n)\cap I_n $ and span $ y $ in $ M $.
Thus  $I_{n+1}\setminus (I_n \vartriangle P_n)$ cannot contain any  edge from $ E(x,n) $.
\end{proof}
\begin{cor}\label{circ subset}
$ I_n\cap  E(x,n)=I_{n+1}\cap E(x,n) $  and  for every $y\in  E(x,n)\setminus I_n $ we have
$ C_M(y,I_n)=C_M(y,I_{n+1})\subseteq E(x,n)$.
\end{cor}
\begin{cor}\label{cor: cocircuit stabil}
For $y\in  E(x,n)\setminus I_n $,  $ y $ has the same out-neighbours in $ D(I_n) $ and in $ D(I_{n+1}) $ and they span $ y $ in
$ N^{*} $.  More concretely:
\[ C_{N^{*}}(y,\overset{\circ}{\mathsf{span}}_{M}(I_{n+1})^{1})=
C_{N^{*}}(y,\overset{\circ}{\mathsf{span}}_{M}(I_{n})^{1}). \]
\end{cor}

Finally,  for $ y\in  E(x,n)$, $ P_n $ does not contain $ y $ or any of its out-neighbours with respect
to $ D(I_n) $  because $ P_n\cap E(x,n)=\varnothing  $. Hence by applying Lemma \ref{l: arc remain lemma} with $P_n,  y$ and
$I_n $   (and then Observation \ref{arc remain fact}) we may conclude that  $ yz\in D(I_{n+1}) $ whenever   $yz\in  D(I_n) $.
This implies $ E(x,n)\subseteq E(x,n+1) $  since reachability from $ x $ is witnessed by the same directed paths.
\end{proof}

Let \[ B:=\bigcup_{m\in \mathbb{N}} \bigcap_{n>m}W\cap I_n. \] We are going to show that $ B $ witnesses that $ W $ is a
wave. Since $ M $ is
finitary the $ M $-independence of the sets  $ I_n\cap W $ implies the $ M $-independence of  $ B $. Similarly $ B^{0} $ is
independent in $ N $ because $ N \upharpoonright E_0 $ is finitary.  Statements (\ref{item stabilize}) and
(\ref{item same circuit}) of Claim \ref{l: stabilazing stuff} ensure $W\subseteq
\mathsf{span}_{M}(B) $, while (\ref{item stabilize}) and (\ref{item same cocircuit}) guarantee $ B^{1} \subseteq
\mathsf{span}_{N^{*}}(W\setminus B) $. The latter means that $ B^{1} $ is independent in $ N.(W^{1}\setminus B^{1}) $.
Suppose for
a contradiction that $ B^{0} $ is not independent in $ N.W^{0} $. Then there exists an $ N $-circuit $ C\subseteq E_0 $ that
meets $ B $
but avoids $ W\setminus B $.  We already know that $ B^{0} $ is $ N $-independent thus $ C $ is not contained in $ B $. Hence
$C\setminus B= C\setminus W\neq\varnothing  $. Let us pick some $ e\in C\cap B$.
Since $ C $ is finite, for every large enough $ n $ we have $ C\cap B\subseteq C\cap I_n $ and  $ I_n $ spans  $ C $ in $ N $ (for
the latter we use
$ X\subseteq W\setminus B $). Applying Corollary \ref{cor: Noutgoing arc}
 with $I_n,  N, C $ and $ e $ tells that  $e\in C_N(f,I_n) $
for
some $ f\in C\setminus W $ whenever $ n $ is large enough.
Then by (\ref{item increasing}) of Claim \ref{l: stabilazing stuff} we can take an $ x\in X $ and an $ n\geq n_x $ such that $ e\in
E(x,n) \cap C_N(f,I_n) $
for some $ f\in C\setminus W $. Then by definition $ f \in E(x,n) \subseteq W $ which contradicts $ f\in C\setminus W $. Thus
$ B^{0} $ is indeed
independent in $ N.W^{0} $ and hence  $ B $ in $ N.W $ as well therefore $ W $ is a wave.

By $ \mathsf{cond}^{+}(M,N) $ we know that $ W $ consists of $ M $-loops and $ r(N.W)=0 $. It implies  $ r(N.X)=0 $
because $ X\subseteq W $ by definition. This means $ X\subseteq \mathsf{span}_N(E\setminus X) $. Since $X\subseteq E_0 $
and $ E_0 $ is the union of the finitary $ N $-components, $ X\subseteq \mathsf{span}_N(E_0\setminus X) $ follows. Thus for
every $ x\in X $
there is a finite $ N $-circuit $ C\subseteq E_0 $ with $ C\cap X=\{ x \} $. The sequence $ (\mathsf{span}_N(I_n)\cap E_0 )$ is
ascending by construction and exhausts $ E_0\setminus X $ by the definition of $ X $. As $ C-x\subseteq E_0\setminus X $ is
finite, this implies that for every large enough $ n $, $ I_n $
spans $ C-x $ in $ N $  and hence spans $ x $ itself as well. But then by the definition of $ X $, we must have
$ X=\varnothing $. Therefore $ (\mathsf{span}_N(I_n)\cap E_0) $    exhausts $ E_0 $  and the proof of
Lemma \ref{l: key-lemma} is complete.
\end{proof}

 \section{An application: Degree-constrained orientations of infinite graphs}\label{s: application}
Matroid intersection is a powerful tool in graph theory and in combinatorial optimization. Our generalization
Theorem \ref{t: main result} extends the scope of  its applicability to infinite
graphs. To illustrate this, let us consider a classical problem in combinatorial optimization.  A graph is given with
degree-constrains and we are looking for either an orientation
that satisfies it or a certain substructure  witnessing the non-existence of such an orientation (see \cite{hakimi1965degrees}).

Let  a  (possibly infinite) graph $ G=(V,E) $ be fixed through this section. We denote the set of edges incident with $ v $ by $
\boldsymbol{\delta(v)} $.
Let $ o: V\rightarrow \mathbb{Z}$ with $ \left|o(n)\right|\leq
d(v) $ for $
v\in V $ which we will threat as `lower bounds' for in-degrees in orientations in the following sense. We say that the orientation $
D $ of $ G $ is
\emph{above} $
o $ at $ v $ if either $
o(v)\geq 0 $ and $ v $ has at least $ o(v) $ ingoing edges in $ D $ or  $ o(v)< 0 $ and all but at most $ -o(v) $ edges in $ \delta(v)
$
are oriented towards $ v $ by $ D $. We say \emph{strictly above} if we forbid equality in the definition. Orientation $ D $ is
above $ o $ if it is above $ o $ at every $ v\in V $. We say that $ D $ is (strictly) bellow $ o $ at $ v $ if the reverse of $ D $ is
(strictly) above $ -o(v) $. Finally, $ D $ is (strictly) bellow $ o $ if the reverse of $ D $ is strictly above $ -o $.

\begin{thm}\label{t: indegree demand}
Let $ G=(V,E) $ be a countable graph and let $ o: V\rightarrow \mathbb{Z} $. If there is no
orientation of $ G $ above $ o $, then there is a  $ V'\subseteq V $ and an orientation $ D $ of $ G $ such that
\begin{itemize}
\item $ D $ is bellow $ o $ at every $ v\in V' $;
\item There exists a $ v\in  V' $ such that $ D $ is strictly bellow $ o $ at $ v $;
\item Every edge between $ V' $ and $ V\setminus V' $ is oriented by $ D $ towards $ V' $.
\end{itemize}
\end{thm}
\begin{proof}
 Without loss of generality we may assume that $ G $ is loopless. We define the digraph $
 \overset{\leftrightarrow}{G}=(V, A) $ by
 replacing each $ e\in E
 $ by back  and forth arcs $ a_{e}, a'_e $ between the
 end-vertices of $ e $. Let $ \delta^{+}(v) $ be the set of the ingoing edges of $ v $ in  $ \overset{\leftrightarrow}{G} $.  For
 $ v\in V $, let $ M_v $ be $
 U_{\delta^{+}(v), o(v)} $ if $
 o(v)\geq 0 $ and $
U_{\delta^{+}(v),-o(v)}^{*} $ if $ o(v)<0 $. We define $ N_e $ to be $ U_{\{a_e, a'_e  \}, 1} $ for $ e\in E $. Let \[
M:=\bigoplus_{v\in V}M_v\text{ and }N:=\bigoplus_{e\in E} N_e. \] Since $ M, N\in
(\mathfrak{F}\oplus \mathfrak{F}^{*})(A) $,  Theorem \ref{t: main result} guarantees that there exists an $ I\in
\mathcal{I}_M\cap
\mathcal{I}_N $ and a partition $ A=A_M\sqcup A_N $  such that $I_M:=I\cap A_M $ spans $ A_M $ in $ M $ and  $
I_N:=I\cap A_N $ spans $ A_N $ in $ N $.  Note that  $ \left|I\cap \{ a_e, a'_e \} \right|\leq 1$ by the $ N $-independence of $ I
$. We define $ D $ by taking the orientation $ a_e $ of $ e $ if $ a_e\in I $ and $ a'_e $ otherwise. Let $ V'' $ consists of those
vertices $ v $ for which $ I_M $ contains a base of $ M_v $ and let $ V':=V\setminus V'' $.
We claim that whenever an edge $ e\in E $ is incident with some $ v\in V' $, then $ I $ contains one of $ a_e $ and $ a'_e $.
Indeed, if $ I $ contains none of them then they cannot be $ N $-spanned by $ I_N $ thus they are $ M $-spanned by $ M $ which
implies that both end-vertices of $ e $ belong to $ V'' $, contradiction. Thus if $ e $ is incident some $ v\in V' $, then all ingoing
arcs of $ v $ in $ D $ must be in $ I $. Then the $ M $-independence of $ I $ ensures that  $ D  $ is bellow $ o $ at $ v $.

Suppose for a contradiction that $ a_e $ is an arc in $ D $ from a $ v\in V' $ to a $ w\in V'' $. As we have already shown, we must
have $ a_e\in I $ . By $ w\in V''
$ we know that $ I_M $ contains a base of
$ M_w $ thus $ a_e\notin I_N $ by the $ M $-independence of $ I $ and therefore $ a_e\in I_M $. But then $ a'_e $ cannot be
spanned by $ I_N $ in $ N $ hence $ a'_e\in \mathsf{span}_M(I_M) $, which means that $ I_M $ contains a base of $ M_v $
contradicting $ v\in V' $. We conclude that all the edges between $ V'' $ and $ V' $ are oriented towards $ V' $ in $ D $. By the
definition of $ V'' $,  $ D $ is above $ o $ at every $ w\in V'' $. If $ D $ is also above $ o $ for every $ v\in V' $, then $ D $
is above $ o $. Otherwise there exists a $v\in  V' $ such that $ D $ is strictly bellow $ o $ at $ v $, but then $ V' $ is as desired.
\end{proof}

Easy calculation shows that if $ G $ is finite, then  the existence of a $ V' $ described in Theorem \ref{t: indegree demand}
implies the non-existence of an orientation above $ o $. Indeed, the total demand by $ o $ on $ V' $ is more than the number of all
the edges that are incident with a vertex in $ V' $. That is why for finite $ G $, ``if'' can be replaced by ``if and only if'' in
Theorem \ref{t: indegree demand}. For an infinite $ G $ it is not always the case. Indeed, let $ G $ be the one-way infinite path $
v_0, v_1,\dots $ and let $ o(v_n)=1 $ for $ n\in \mathbb{N} $. Then orienting edge $ \{ v_{n}, v_{n+1} \} $ towards $ v_{n+1}
$ for each $ n\in \mathbb{N} $ and taking $ V':=V $ satisfies the three points in Theorem \ref{t: indegree demand}. However,
taking the opposite orientation is above $ o $.

A natural next step is to  introduce upper bounds $ p:V\rightarrow \mathbb{Z}  $ beside the lower bounds
$ o:V\rightarrow \mathbb{Z}  $. To avoid trivial obstructions we assume that $ o $ and $ p $ are \emph{consistent} which means
that for every $ v\in V $ there is an orientation $ D_v $
which is above $ o $ at bellow $ p $ at $ v $.

\begin{quest}
Let $ G $ be a countable graph and let $ o, p: V\rightarrow \mathbb{Z} $ be a consistent pair of bounding functions.
Suppose that there are orientations $ D_o $ and $ D_p $  that are above $ o $ and bellow $ p $ respectively. Is there always a
single
orientation $ D $ which is above $ o $ and bellow $ p $?
\end{quest}

The positive answer for finite graphs is not too hard to prove,  as far we know its first appearance in the literature is
\cite{frank1978orient}.

\begin{bibdiv}
\begin{biblist}

\bib{aharoni1984konig}{article}{
      author={Aharoni, Ron},
       title={K{\"o}nig's duality theorem for infinite bipartite graphs},
        date={1984},
     journal={Journal of the London Mathematical Society},
      volume={2},
      number={1},
       pages={1\ndash 12},
}

\bib{aharoni2009menger}{article}{
      author={Aharoni, Ron},
      author={Berger, Eli},
       title={Menger’s theorem for infinite graphs},
        date={2009},
     journal={Inventiones mathematicae},
      volume={176},
      number={1},
       pages={1\ndash 62},
}

\bib{aharoni1983general}{article}{
      author={Aharoni, Ron},
      author={Nash-Williams, Crispin},
      author={Shelah, Saharon},
       title={A general criterion for the existence of transversals},
        date={1983},
     journal={Proceedings of the London Mathematical Society},
      volume={3},
      number={1},
       pages={43\ndash 68},
}

\bib{aharoni1984another}{article}{
      author={Aharoni, Ron},
      author={Nash-Williams, Crispin},
      author={Shelah, Saharon},
       title={Another form of a criterion for the existence of transversals},
        date={1984},
     journal={Journal of the London Mathematical Society},
      volume={2},
      number={2},
       pages={193\ndash 203},
}

\bib{aharoni1998intersection}{article}{
      author={Aharoni, Ron},
      author={Ziv, Ran},
       title={The intersection of two infinite matroids},
        date={1998},
     journal={Journal of the London Mathematical Society},
      volume={58},
      number={03},
       pages={513\ndash 525},
}

\bib{aigner2018intersection}{article}{
      author={Aigner-Horev, Elad},
      author={Carmesin, Johannes},
      author={Fr{\"o}hlich, Jan-Oliver},
       title={On the intersection of infinite matroids},
        date={2018},
     journal={Discrete Mathematics},
      volume={341},
      number={6},
       pages={1582\ndash 1596},
}

\bib{nathanhabil}{thesis}{
      author={Bowler, Nathan},
       title={Infinite matroids},
        type={Habilitation Thesis},
        date={2014},
  note={\url{https://www.math.uni-hamburg.de/spag/dm/papers/Bowler\_Habil.pdf}},
}

\bib{bowler2015matroid}{article}{
      author={Bowler, Nathan},
      author={Carmesin, Johannes},
       title={Matroid intersection, base packing and base covering for infinite
  matroids},
        date={2015},
     journal={Combinatorica},
      volume={35},
      number={2},
       pages={153\ndash 180},
}

\bib{bowler2016self}{article}{
      author={Bowler, Nathan},
      author={Geschke, Stefan},
       title={Self-dual uniform matroids on infinite sets},
        date={2016},
     journal={Proceedings of the American Mathematical Society},
      volume={144},
      number={2},
       pages={459\ndash 471},
}

\bib{bruhn2013axioms}{article}{
      author={Bruhn, Henning},
      author={Diestel, Reinhard},
      author={Kriesell, Matthias},
      author={Pendavingh, Rudi},
      author={Wollan, Paul},
       title={Axioms for infinite matroids},
        date={2013},
     journal={Advances in Mathematics},
      volume={239},
       pages={18\ndash 46},
}

\bib{edmonds1968matroid}{article}{
      author={Edmonds, Jack},
       title={Matroid partition},
        date={1968},
     journal={Mathematics of the Decision Sciences},
      volume={11},
       pages={335\ndash 345},
}

\bib{edmonds2003submodular}{incollection}{
      author={Edmonds, Jack},
       title={Submodular functions, matroids, and certain polyhedra},
        date={2003},
   booktitle={Combinatorial optimization—eureka, you shrink!},
   publisher={Springer},
       pages={11\ndash 26},
}

\bib{erde2019base}{article}{
      author={Erde, Joshua},
      author={Gollin, J.~Pascal},
      author={Jo{\'o}, Attila},
      author={Knappe, Paul},
      author={Pitz, Max},
       title={Base partition for mixed families of finitary and cofinitary
  matroids},
        date={2021},
        ISSN={1439-6912},
     journal={Combinatorica},
      volume={41},
      number={1},
       pages={31\ndash 52},
         url={https://doi.org/10.1007/s00493-020-4422-4},
}

\bib{frank1978orient}{article}{
      author={Frank, Andr{\'a}s},
       title={How to orient the edges of a graph?},
        date={1978},
     journal={Combinatorics},
       pages={353\ndash 364},
}

\bib{frank2011connections}{book}{
      author={Frank, Andr{\'a}s},
       title={Connections in combinatorial optimization},
   publisher={OUP Oxford},
        date={2011},
      volume={38},
}

\bib{ghaderi2017}{thesis}{
      author={Ghaderi, Shadisadat},
       title={On the matroid intersection conjecture},
        type={PhD. Thesis},
        date={2017},
}

\bib{hakimi1965degrees}{article}{
      author={Hakimi, S~Louis},
       title={On the degrees of the vertices of a directed graph},
        date={1965},
     journal={Journal of the Franklin Institute},
      volume={279},
      number={4},
       pages={290\ndash 308},
}

\bib{higgs1969equicardinality}{article}{
      author={Higgs, DA},
       title={Equicardinality of bases in {B}-matroids},
        date={1969},
     journal={Can. Math. Bull},
      volume={12},
       pages={861\ndash 862},
}

\bib{higgs1969matroids}{inproceedings}{
      author={Higgs, Denis~Arthur},
       title={Matroids and duality},
        date={1969},
   booktitle={Colloquium mathematicum},
      volume={2},
       pages={215\ndash 220},
}

\bib{joó2020intersection}{misc}{
      author={Joó, Attila},
       title={Intersection of a partitional and a general infinite matroid},
         how={arXiv},
        date={2020},
        note={\url{https://arxiv.org/abs/2009.07205}},
}

\bib{joo2020MIC}{article}{
      author={Jo{\'o}, Attila},
       title={Proof of {N}ash-{W}illiams' intersection conjecture for countable
  matroids},
        date={2021},
        ISSN={0001-8708},
     journal={Advances in Mathematics},
      volume={380},
       pages={107608},
}

\bib{oxley1978infinite}{article}{
      author={Oxley, James},
       title={Infinite matroids},
        date={1978},
     journal={Proc. London Math. Soc},
      volume={37},
      number={3},
       pages={259\ndash 272},
}

\bib{oxley1992infinite}{article}{
      author={Oxley, James},
       title={Infinite matroids},
        date={1992},
     journal={Matroid applications},
      volume={40},
       pages={73\ndash 90},
}

\bib{rado1966abstract}{inproceedings}{
      author={Rado, Richard},
       title={Abstract linear dependence},
organization={Institute of Mathematics Polish Academy of Sciences},
        date={1966},
   booktitle={Colloquium mathematicum},
      volume={14},
       pages={257\ndash 264},
}

\end{biblist}
\end{bibdiv}

\end{document}

\section{Outlook: Multi-Matroid Intersection by Aharoni}
For $ k\geq 2 $, a  $k $\emph{-partite hypergraph}  is a $ H=(V_1,\dots, V_k; E) $ where the sets $ V_i $ are pairwise disjoint
and each
edge
$ e\in E $ is a set of size $ k $ containing one element from each $ V_i $. A set of pairwise disjoint edges is called a
\emph{matching} and
a set of vertices meeting every edge is a \emph{vertex cover}. The size of a largest matching and a smallest vertex cover denoted
by $ \boldsymbol{\nu(H)} $ and $ \boldsymbol{\tau(H)} $ respectively.
\begin{conj}[Ryser’s Conjecture]\label{conj: Rys}
For every finite $ k $-partite hypergraph $ H $,  $ \tau(H)\leq (k-1)\nu(H) $.
\end{conj}
Aharoni settled the question affirmatively for $ k=3 $ in \cite{aharoni2001ryser} and conjectured the following structural
generalization which are equally
meaningful for infinite $ k $-partite hypergraphs:
\begin{conj}\label{conj: str Rys}
For every hypergraph  $ H=(V_1,\dots, V_k; E) $ there are matchings $ I_1, \dots, I_{k-1} $ such that one can choose
for every $  1\leq i\leq k   $  and $ e\in I_i  $ a vertex $ v_{i,e}\in e $ such that the (multi)set
$  \{v_{i,e}: 1\leq i\leq k-1 \wedge e\in I_i  \} $ is a vertex cover.
\end{conj}
\noindent For a finite $ H $  the Pigeonhole principle ensues that one of the matchings $ I_i $ must have at least $
\frac{\tau(H)}{k-1} $ elements,
hence this is indeed a generalization of Ryser's Conjecture \ref{conj: Rys}.

The matchings of a $ k $-partite hypergraph  can be considered as common
independent sets of the partition matroids $ M_1,\dots, M_k $ on $ E $ where $I\in \mathcal{I}_{M_i} $ iff no distinct $ e,f\in I
$ have a common vertex in $ V_i $. This motivated the following matroidal generalization:
\begin{conj}[Aharoni and Berger]
Assume that $ M_1,\dots, M_k $ are matroids on the finite edge
set $ E $ where $ 2\leq k\in \mathbb{N} $. Then they
admit common independent set of size at least
\[ \frac{1}{k-1} \min \left\lbrace  \sum_{i=1}^{k}r_i(X_i): E= \bigsqcup_{i=1}^{k} X_i. \right\rbrace   \]
\end{conj}

The case $ k=3 $ was solved by Aharoni and Berger in \cite{aharoni2006intersection}.

\begin{conj}[Multi-Matroid Intersection Conjecture by Aharoni]
Assume that $ M_1,\dots, M_k $ are arbitrary/finitary/finite matroids on $ E $ where $ 2\leq k\in \mathbb{N} $. Then they
admit common independent sets $ I_1,\dots, I_{k-1} $ such that for
each $ 1\leq i\leq k-1 $ there is a $ k $-partition $ I_i=\bigsqcup_{j=1}^{k}I_i^{j}  $  for which
\[ \bigcup_{j=1}^{k}\mathsf{span}_{M_j}\left( \bigcup_{i=1}^{k-1}I_{i}^{j}\right) =E. \]
\end{conj}

For $ k=2 $, we get the structural formulation of the Matroid Intersection Theorem by Edmonds if we consider finite matroids,
the Matroid Intersection Conjecture by Nash-Williams if the matroids are assumed to be finitary and the Generalized Matroid
Intersection Conjecture if the matroids are arbitrary.


\section{\bf{I.} Sample structure for the record-quality two-dimensional electron systems}

\begin{figure*}[h]
\centering
    \includegraphics[width=.80\textwidth]{Supplement_R1_Page_2.png}

 \caption{\label{figS1}Sample structure for the record-quality GaAs two-dimensional electron systems whose data are shown in the main text. The spacer layer thickness is varied to obtain the desired sample density, and the quantum well width is chosen accordingly to prevent second-subband occupation.}
\end{figure*}

All record-quality samples follow the structure shown in Fig. S1. Standard modulation-doped structures are $\delta$-doped in AlGaAs while the doping-well structures (DWSs) use the doping scheme described in the right panels of the figure. A 12\%/24\% stepped-barrier structure is implemented to reduce the Al composition of the barrier directly in contact with the main quantum well. The thickness of each of the barrier layers is controlled so that no parallel channel forms at the 12\%/24\% barrier interface, and the total spacer thickness is varied to attain the desired sample density. The well width of the quantum well is varied for each sample so that there is no second-subband occupation. Specific structural parameters are summarized in Table S1 for some representative samples.

\newpage

\begin{table}[h]
\begin{center}
\begin{tabular}{ |c|c|c|c|c| }

\hline
 $n$ ($\times10^{11}$ /cm$^2$)& $\mu$ ($\times10^6$ cm$^{2}$/Vs) & $s_1$ (nm) &  $s_2$ (nm) & $w$ (nm)  \\
\hline
\hline
2.06 & 44.0 & 68.2 & 60.0 & 34.0 \\
\hline
1.15 & 41.5 & 114 & 100 & 45.3 \\
\hline
1.00 & 36.0 & 136 & 120 & 50.0 \\
\hline
0.71 & 29.0 & 195 & 171 & 58.5 \\
\hline
\end{tabular}

\caption{Structural parameters of some representative record-quality doping-well-structure samples whose data are shown in the main text; for definitions of $s_1$, $s_2$, and $w$, see the sample structure in Fig. S1.} \label{tab:table1}

\end{center}
\end{table}

\newpage

\section{\bf{II.} Measurement of energy gap for the $\nu=5/2$ fractional quantum Hall state}

\begin{figure}[h]
\centering
    \includegraphics[width=.50\textwidth]{Supplement_R3_Page_2.png}

 \caption{\label{figS2} Arrhenius plot of resistance at $\nu=5/2$. The sample density is $n\simeq1.0\times10^{11}$/cm$^2$. }
\end{figure}

Magnetoresistance traces were taken at a sweep rate of 0.1 T/hour in the vicinity of the $\nu=5/2$ fractional quantum Hall state to precisely determine the magnetic field position of the $R_{xx}$ minimum at base temperature. Then the energy gap $^{5/2}\Delta$ was measured by raising the temperature of the sample while monitoring the sample resistance at the magnetic field corresponding to $\nu=5/2$. The data are shown in the Arrhenius plot of Fig. S2. A line corresponding to $R_{xx}\sim e^{-^{5/2}\Delta/{2k_BT}}$ was fitted to the data plotted in Fig. S2 (shown in red), yielding an energy gap of $^{5/2}\Delta=820$ mK. Here $k_B$ is the Boltzmann constant.

\newpage

\end{document}