Is there a family $\{A\_\alpha:\alpha<2^{\omega\_1}\}\subset [\omega\_1]^{\omega\_1}$ in ZFC such that for each countable set $I\subset 2^{\omega\_1}$ and $\alpha\in 2^{\omega\_1}\setminus I$ we have
$$A\_\alpha\not\subset\bigcup\{A\_\beta:\beta\in I\}.$$


Of course, we can find such a family of size $\omega\_2$ instead of $2^{\omega\_1}$.



A question on subsets of $\omega_1$
Today I could prove that the existence of such family is not provable from ZFC. 


**Theorem.**
If GCH holds in $V$, and $\mu\ge \omega\_3$ then in $V^{Fn(\mu,2)}$ the following holds:
*If $\{A\_i:i<{\omega}\_3\}\subset [{\omega}\_1]^{{\omega}\_1}$, then there is
 $I\in [\omega\_3]^\omega$ and $\alpha\in \omega\_3\setminus I $ such that 
 $$A\_{\alpha}\subset \bigcup\_{i\in I}A\_i.$$*


**Proof.**
An ${Fn(I,2)}$-name $\dot B$ of a subset of $\omega\_1$ is called 
 *nice* if for each ${\nu}<\omega\_1$ there is an antichain 
$B\_{\nu}\subset {Fn(I,2)}$ such that 
$$
\dot B=\{\langle p,\check{{\nu}}\rangle:{\nu}\in {\omega\_1}\land p\in B\_{\nu}\}=
\bigcup\{B\_{\nu}\times\{\check{{\nu}}\}:{\nu}\in {\omega\_1}\}.
$$
We let $supp(\dot B)=\bigcup\{dom(p):p\in\bigcup\limits\_{{\nu}<{\omega\_1}}B\_{\nu}\}$.


It is well-known that every set of ordinals in $V[G]$ has a nice name in $V$.


If ${\varphi}$ is a bijection between two sets $I$ and $J$ then
${\varphi}$ lifts to a natural isomorphism between ${Fn(I,2)}$ and
${Fn(I,2)}$, which will be also denoted by ${\varphi}$, as follows: 
for $p\in {Fn(I,2)}$ let $dom({\varphi}(p))={\varphi}''dom(p)$ and
${\varphi}(p)({\varphi}({\xi}))=p({\xi})$. Moreover ${\varphi}$ also
generates a bijection between the nice ${Fn(I,2)}$-names and the nice
$Fn(J,2)$-names: 
if $\dot A$ is a nice $Fn(I,2)$-name then let
${\varphi}(\dot A )=\{\langle{\varphi}(p),\hat{{\xi}}\>:\langle p,\hat{{\xi}}\rangle \in\dot B\}$.


If $I$ and $J$ are sets of ordinals with the same order type then
 ${\varphi}\_{I,J}$ is the natural order-preserving
bijection from $I$ onto $J$.


For each $i<{\omega}\_3$ pick a nice $Fn(\mu,2)$-name $\dot A\_i$ 
for $A\_i$ in $V$. Then $S\_i=supp(A\_i)\in [\mu]^{\le \omega\_1}$.


Since $2^{\omega\_1}=\omega\_2$ in $V$ there is a subset 
$K\in [\omega\_3]^{\omega\_3}$ such that 


(1) $\{S\_i:i\in K\}$ forms a $\Delta$-system with kernel $S$, 


(2) every element of the family $\{S\_i:i\in K\}$ has the same
 order type 


(3) $\varphi\_{S\_i,S\_j}$ is the identity on $S$ for all
 $i,j\in K$. 


(4) $\varphi\_{S\_i,S\_J}(\dot A\_i)=\dot A\_j$ for
 $i,j\in K$.


Let $I\in [K]^\omega$ and $\alpha \in K\setminus I$.
Assume on the contrary that $p\in Fn(\mu,2)$ and $\zeta<\omega\_1$ such that 
$$
p\Vdash \check \zeta\in \dot A\_{\alpha}\setminus \bigcup\_{i\in I}\dot A\_i. 
$$
Pick $\beta \in I$ such that $dom(p)\cap (S\_\beta\setminus S)=\emptyset.$
Let 
$$
r=p\cup \varphi\_{S\_\alpha, S\_\beta}(p\restriction S\_\alpha).
$$
Since $dom(p)\cap (S\_\beta\setminus S)=\emptyset$, $r$ is a condition.
Moreover, $p\Vdash \check \zeta\in \dot A\_{\alpha}$,
so $p\restriction S\_\alpha \Vdash \check \zeta\in \dot A\_{\alpha}$,
and so $\varphi\_{S\_\alpha, S\_\beta}(p\restriction S\_\alpha )\Vdash 
\check \zeta\in \dot A\_{\beta}$. 


Thus $r\Vdash \zeta\in \bigcup\_{i\in I}\dot A\_i.$ Contradiction.


