Ramsey numbers and the size of graphs

TL;DR

This paper investigates how the Ramsey number r(K_s, G) depends on the size of G, providing new lower bounds that improve previous results and are tight for specific cases.

Contribution

It establishes a new lower bound for r(K_s, G) based on the number of edges in G, improving earlier bounds and analyzing its maximum as a function of G's size.

Findings

New lower bound for r(K_s, G) in terms of edges m

Improved bounds are tight for s=3

Analysis of maximum r(K_s, G) as a function of m

Abstract

For two graph H and G, the Ramsey number r(H, G) is the smallest positive integer n such that every red-blue edge coloring of the complete graph K_n on n vertices contains either a red copy of H or a blue copy of G. Motivated by questions of Erdos and Harary, in this note we study how the Ramsey number r(K_s, G) depends on the size of the graph G. For s \geq 3, we prove that for every G with m edges, r(K_s,G) \geq c (m/\log m)^{\frac{s+1}{s+3}} for some positive constant c depending only on s. This lower bound improves an earlier result of Erdos, Faudree, Rousseau, and Schelp, and is tight up to a polylogarithmic factor when s=3. We also study the maximum value of r(K_s,G) as a function of m.