On Induced Subgraphs of Finite Graphs not Containing Large Empty and Complete Subgraphs

Abstract: In their celebrated paper [Ramsey-Type Theorems, Discrete Appl. Math. 25 (1989) 37-52], Erd\H{o}s and Hajnal asked the following: is it true, that for any finite graph H there exists a constant c(H) such that for any finite graph G, if G does not contain complete or empty induced subgraphs of size at least |V(G)|^c(H), then H can be isomorphically embedded into G ? The positive answer has become known as the Erd\H{o}s-Hajnal conjecture. In Theorem 3.20 of the present paper we settle this conjecture in the affirmative. To do so, we are studying here the fine structure of ultraproducts of finite sets, so our investigations have a model theoretic character.