On bounded degree graphs with large size-Ramsey numbers

TL;DR

This paper improves lower bounds on the size-Ramsey number for bounded degree graphs, showing that certain graphs with degree at most three have size-Ramsey numbers growing faster than previously known, specifically at least on the order of n times an exponential of the square root of log n.

Contribution

The authors present a modified construction that establishes a new lower bound on the size-Ramsey number for graphs with maximum degree three, surpassing earlier bounds involving logarithmic factors.

Findings

Established a lower bound of  n  e^{c \u221a{\u221a{ ext{log} n}}} for the size-Ramsey number of certain bounded degree graphs.

Demonstrated that the size-Ramsey number can grow significantly faster than previously shown for graphs with maximum degree three.

Provided a construction that improves the understanding of the extremal behavior of size-Ramsey numbers for bounded degree graphs.

Abstract

The size-Ramsey number $\hat r(G')$ of a graph $G'$ is defined as the smallest integer $m$ so that there exists a graph $G$ with $m$ edges such that every $2$-coloring of the edges of $G$ contains a monochromatic copy of $G'$. Answering a question of Beck, Rodl and Szemeredi showed that for every $n\geq 1$ there exists a graph $G'$ on $n$ vertices each of degree at most three, with the size-Ramsey number at least $cn\log^{\frac{1}{60}}n$ for a universal constant $c>0$. In this note we show that a modification of Rodl and Szemeredi's construction leads to a bound $\hat r(G')\geq cn\,\exp(c\sqrt{\log n})$.