# 2014 seminar talk: Y-c.c. and Y-proper forcing notions

Talk held by David Chodounsky (Institute of Mathematics of the Czech Academy of Sciences) at the KGRC seminar on 2014-11-27.

### Abstract

I will introduce a new type of properties of c.c.c. and proper forcing notions, of which the Y-c.c. and Y-proper are the two most prominent examples. The Y-c.c. is an intermediate property between σ-centered and c.c.c., and Y-proper is intermediate between strongly proper and proper. These properties have interesting consequences, let us mention some: adding an unbounded real, preserving uncountable chromatic number of open graphs, not adding random reals, and not adding uncountable branches into trees. Moreover, partial orders with these properties appear naturally and many classical forcing notions do poses them. These properties behave also reasonably with respect to forcing iteration, and it is possible to get corresponding forcing axioms using posets in respective classes. The talk will focus on introducing these properties and proving basic fact about them.

This is a joint work with Jindra Zapletal.