Is the following claim correct (Chapter 13 before Theorem 87 of Todorcevic's book: Notes on forcing axioms):
Let $\alpha$ be an infinite countable indecomposable ordinal and $U$ be an uniform ultrafilter on $\alpha$ (namely elements in $U$ have order type $\alpha$). Then for any collection $\{B\_i\in U: i<\mathfrak{m}\}$ it is true that there exists $B\subset \alpha$ of order type $\alpha$ such that $B-B\_i$ is finite for all $i<\mathfrak{m}$. Here $\mathfrak{m}$ is the least cardinal such that Martin's Axiom holds below $\mathfrak{m}$ (i.e. meeting any $\beta$ many dense sets for any $\beta<\mathfrak{m}$).


In fact, it will be interesting to know if this is true at all for countable collection $\{B\_i: i<\omega\}$. 


It is proved there that the statement is true if we relax the conclusion asking only $B-B\_i$ to be bounded in $B$. Any thoughts?



Indecomposable ordinals and pseudointersection
I believe the claim is wrong: 


If the claim is right I claim I can show $\alpha\to (\alpha)^2\_2$ which is obviously wrong for countable ordinal $\alpha\geq \omega+2$. 


Given a coloring $f: [\alpha]^2\to 2$, for each $\beta\in \alpha$, let $A\_\beta^i=\{\gamma<\alpha: f(\beta,\gamma)=i\}$ for $i<2$. Let $g: \alpha\to 2$ be such that $A\_{\beta}^{g(\beta)}\in U$. 
Find $i<2$ and $B'\in U$ such that for each $\beta\in B'$, $g(\beta)=i$.
$\{A\_\beta^{i}: \beta\in B'\}$ is a countable collection of elements in $U$, hence there exists a pseudointersection $B\subset B'$ such that $B$ has order type $\alpha$ and $B-A\_{\beta}^{i}$ is finite for any $\beta \in B'$.


Now decompose $B$ into $\alpha$ increasing intervals each of which has order type $\omega$, say $\{I\_\xi: \xi<\alpha\}$. We inductively in $\omega$-stages pick a monochromatic subset of order type $\alpha$ such that $I\_\xi$ contributes exactly one element. Fix a bijection $h: \omega\to \alpha$. Suppose $\alpha\_j$ have been picked for $j<k$. We will pick $\alpha\_k$ in $I\_{h(k)}$. Since $\bigcup\_{j<k} B-A\_{\alpha\_j}^i$ is finite, there exists $\alpha\_{k+1}\in I\_{h(k)}\cap \bigcap\_{j<k} A\_{\alpha\_j}^i$. Then eventually we will define a monochromatic subset of order type $\alpha$ (with constant color $i$).


