TITLE: "generic" in elementary submodels
QUESTION [10 upvotes]: Suppose you have a Suslin tree $T$ and you have a countable elementary submodel $M$ containing the usual "enough stuff" (including $T$). A comment in Todorcevic's Partition Problems in Topology indicates that a branch of $T$ of height $M \cap \omega_1$ will be generic, i.e., will meet every dense set of $T$ that's an element of $M$. Why? And is the result more general than for Suslin trees? I.e., does it use the fact that their branches are countable (so $M$ knows of no branch of height $M \cap \omega_1$)? Or does it hold for more general partial orders?

REPLY [5 votes]: In answer to Judy's second question, the fact that branches are all countable is not enough to get genericity of any branch of the same height as the elementary submodel. For example, consider the classic Aronszajn tree consisting of one-to-one functions from the ordinals to the integers. For any integer k the set of all functions with k in their range is an antichain. However, it is easy to construct a branch going to the top of an elementary submodel without k in its range, hence missing this definable antichain.