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?