Induced Ramsey-type theorems

TL;DR

This paper introduces a unified method for proving induced Ramsey theorems in graphs, improving bounds and extending previous results by replacing complex tools with a simpler lemma.

Contribution

A new unified approach that simplifies proofs of induced Ramsey theorems and yields better bounds and explicit constructions for induced Ramsey numbers.

Findings

Improved bounds on induced Ramsey numbers.

Extension of classical theorems to broader graph classes.

Explicit constructions for pseudo-random graphs with strong induced Ramsey properties.

Abstract

We present a unified approach to proving Ramsey-type theorems for graphs with a forbidden induced subgraph which can be used to extend and improve the earlier results of Rodl, Erdos-Hajnal, Promel-Rodl, Nikiforov, Chung-Graham, and Luczak-Rodl. The proofs are based on a simple lemma (generalizing one by Graham, Rodl, and Rucinski) that can be used as a replacement for Szemeredi's regularity lemma, thereby giving much better bounds. The same approach can be also used to show that pseudo-random graphs have strong induced Ramsey properties. This leads to explicit constructions for upper bounds on various induced Ramsey numbers.