Around a conjecture of ErdH{o}s on graph Ramsey numbers

TL;DR

This paper discusses an extension of Sudakov's proof of Erdős's conjecture on graph Ramsey numbers, providing new bounds and corollaries related to graphs with specific properties.

Contribution

The paper extends Sudakov's proof of Erdős's conjecture, offering new bounds on Ramsey numbers and deriving interesting corollaries for graphs with no isolated vertices.

Findings

Confirmed Erdős's conjecture with extended bounds

Derived new corollaries for graphs with no isolated vertices

Provided a generalized approach based on Sudakov's ideas

Abstract

For given graphs G1 and G2 the Ramsey number R(G1,G2), is the smallest positive integer n such that each blue-red edge coloring of the complete graph Kn contains a blue copy of G1 or a red copy of G2. In 1983, Erdos conjectured that there is an absolute constant c such that R(G) = R(G,G) < 2c p m for any graph G with m edges and no isolated vertices. Recently this conjecture was proved by B. Sudakov. In this note, using the Sudakovs ideas we give an extension of his result and some interesting corollaries.