On Ramsey size-linear graphs and related questions

TL;DR

This paper investigates Ramsey numbers for fixed and large graphs, extending previous work on size-linear graphs, and proves new bounds for specific graph subdivisions and related conjectures.

Contribution

It extends existing results on Ramsey size-linear graphs, proving new bounds for subdivisions of K4 and advancing conjectures on Ramsey numbers for connected graphs.

Findings

For subdivisions of K4 with at least 6 vertices, R(H,F) = O(v(F) + e(F)).

Proved the case k=3 for a conjecture relating Ramsey numbers to graph parameters.

Extended the class of graphs with known size-linear Ramsey bounds.

Abstract

In this paper we prove several results on Ramsey numbers $R(H,F)$ for a fixed graph $H$ and a large graph $F$, in particular for $F = K_n$. These results extend earlier work of Erd\H{o}s, Faudree, Rousseau and Schelp and of Balister, Schelp and Simonovits on so-called Ramsey size-linear graphs. Among others, we show that if $H$ is a subdivision of $K_4$ with at least $6$ vertices, then $R(H,F) = O(v(F) + e(F))$ for every graph $F$. We also conjecture that if $H$ is a connected graph with $e(H) - v(H) \leq \binom{k+1}{2} - 2$, then $R(H,K_n) = O(n^k)$. The case $k=2$ was proved by Erd\H{o}s, Faudree, Rousseau and Schelp. We prove the case $k=3$.