Borel chromatic numbers in models of set theory

Abstract

In this work, we study the behavior of definable graphs on Polish spaces in various models of set theory. More specifically, we investigate their Borel chromatic numbers, one of the so-called cardinal characteristics of the continuum. We show that the statement “the Borel chromatic number of a graph is bounded by the continuum of the ground model” may be forced, depending on (1) the topology of the space of vertices; (2) the complexity of the graph (e.g., analytic, closed etc); and on (3) some suitable notion of “smallness” which may be satisfied for the graph (e.g., local countability, the inexistence of perfect cliques etc). For that, we use countable support iterations of Axiom A forcing notions. Furthermore, from the results of Chapter 3 we are also able to solve a relatively old problem about regularity properties, showing that Silver and Laver measurability may be separated on the second level of the projective hierarchy. The content of Chapter 2 is a joint work with Stefan Geschke; and the content of Chapter 3 is a joint work with Raiean Banerjee.