Erdős–Hajnal and VC-dimension

A 1977 conjecture of Erdős and Hajnal asserts that for every hereditary class of graphs not containing all graphs, every graph in the class has a polynomial-sized clique or stable set. Fox, Pach, and Suk and independently Chernikov, Starchenko, and Thomas asked whether this conjecture holds for every class of graphs of bounded VC-dimension. In joint work with Alex Scott and Paul Seymour, we resolved this question in the affirmative. The talk will introduce the Erdős–Hajnal conjecture and discuss some ideas behind the proof of the bounded VC-dimension case.