# Blog Archives

# Topic Archive: tree property

# Reflection of stationary sets, tree property and square

One of the most fruitful research area in set theory is the study of the so-called Reflections Principles. Roughly speaking, a reflection principle is a combinatorial statement of the following form: given a structure S (e.g. stationary sets, tree, graphs, groups …) and a property P of the structure, the principle establishes that there exists a smaller substructure of S that satisfies the same property P. There is a tension between large cardinals axioms and the axiom of constructibility V=L at the level of reflection: on the one hand, large cardinals typically imply reflection properties, on the other hand L satisfies the square principles which are anti-reflection properties. Two particular cases of reflection received special attention, the reflection of stationary sets and the tree property. We will discuss the interactions between these principles and a version of the square due to Todorcevic. This is a joint work with Menachem Magidor and Yair Hayut.