# “generic” in elementary submodels

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?

• Hi Judy. Welcome! – Andrés E. Caicedo Mar 22 '12 at 20:56
• Hi Professor Roitman. It is so cool to see you on mathoverflow. I like your question. – user10290 Mar 25 '12 at 19:06

In your case, you have the countable elementary substructure $M$, and want to know about $M$-genericity. But the same observation works. Every antichain in $M$ is refined by a level of the tree in $M$, and since any branch of the tree of height above $\omega_1\cap M$ goes through all such levels, it will therefore meet every such maximal antichain in $M$, and consequently it will be $M$-generic
One key fact for this argument was the fact that the forcing was ccc---the antichains were countable---but a powerful generalization of similar ideas is the notion of proper forcing, which can be defined in terms of finding conditions that force $M$-genericity for suitable countable elementary submodels.