Asymptotics of Ramsey numbers of double stars

TL;DR

This paper investigates the Ramsey numbers of double star graphs, extending previous results with computational methods and identifying ranges where the conjecture holds or fails, also addressing a question from 1982.

Contribution

The authors extend known bounds on Ramsey numbers of double stars using flag algebra computations and identify specific parameter ranges where the conjecture does not hold.

Findings

Confirmed the conjecture for certain parameter ranges using computational methods.

Disproved the conjecture for other parameter ranges, providing counterexamples.

Answered a long-standing question from 1982 with new examples.

Abstract

A double star $S(n,m)$ is the graph obtained by joining the center of a star with $n$ leaves to a center of a star with $m$ leaves by an edge. Let $r(S(n,m))$ denote the Ramsey number of the double star $S(n,m)$. In 1979 Grossman, Harary and Klawe have shown that $$r(S(n,m)) = \max\{n+2m+2,2n+2\}$$ for $3 \leq m \leq n\leq \sqrt{2}m$ and $3m \leq n$. They conjectured that equality holds for all $m,n \geq 3$. Using a flag algebra computation, we extend their result showing that $r(S(n,m))\leq n+ 2m + 2$ for $m \leq n \leq 1.699m$. On the other hand, we show that the conjecture fails for $\frac{7}{4}m +o(m)\leq n \leq \frac{105}{41}m-o(m)$. Our examples additionally give a negative answer to a question of Erd\H{o}s, Faudree, Rousseau and Schelp from 1982.