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

TL;DR

This paper proves the Erd ext{"o}s-Hajnal conjecture, showing that for any finite graph, large empty or complete induced subgraphs imply the presence of any smaller graph, using model-theoretic methods involving ultraproducts.

Contribution

It provides a proof of the Erd ext{"o}s-Hajnal conjecture by analyzing ultraproducts of finite sets with a model-theoretic approach.

Findings

Confirmed the Erd ext{"o}s-Hajnal conjecture.

Established a link between graph structure and ultraproduct analysis.

Used model theory to solve a longstanding combinatorial problem.

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.