Why Y-c.c

TL;DR

This paper introduces new iterable properties of c.c.c. and proper forcing notions, specifically Y-c.c. and Y-properness, exploring their implications for partition forcing and open graph anticliques, and demonstrating consistency results via Neeman's method.

Contribution

It presents novel iterable properties of c.c.c. and proper forcing notions, including Y-c.c. and Y-properness, with applications to partition forcing and consistency proofs.

Findings

Y-c.c. and Y-properness have significant consequences for partition-type forcings.

Using Neeman's side condition method, the paper proves consistency results for PFA variations.

The properties influence the structure of anticliques in open graphs.

Abstract

We outline a portfolio of novel iterable properties of c.c.c. and proper forcing notions and study its most important instantiations, Y-c.c. and Y-properness. These properties have interesting consequences for partition-type forcings and anticliques in open graphs. Using Neeman's side condition method it is possible to obtain PFA variations and prove consistency results for them.