Multicolor Ramsey Number for Double Stars

TL;DR

This paper determines the exact multicolor Ramsey number for double stars when the number of colors is odd and large enough, advancing understanding of these numbers and their relation to list Ramsey numbers.

Contribution

It provides new exact formulas for the Ramsey numbers of double stars in certain cases, addressing a gap in the knowledge of these classical graph invariants.

Findings

r(S(n,m);k)=kn+m+2 for odd k and large n

r(S^m_n;k)=k(n-1)+m+2 under similar conditions

Initial insights into the relationship between Ramsey and list Ramsey numbers for double stars

Abstract

For a graph $H$ and an integer $k\ge1$, let $r(H;k)$ and $r_\ell(H;k)$ denote the $k$-color Ramsey number and list Ramsey number of $H$, respectively. Alon, Buci\'c, Kalvari, Kuperwasser and Szab\'o in 2021 initiated the systematic study of list Ramsey numbers of graphs and hypergraphs, and conjectured that $ r(K_{1,n};k)$ and $r_\ell(K_{1,n};k)$ are always equal. Motivated by their work, we study the $k$-color Ramsey number for double stars $S(n,m)$, where $n\ge m\ge1$. To the best of our knowledge, little is known on the exact value of $r(S(n,m);k)$ when $k\ge3$. A classic result of Erd\H{o}s and Graham from 1975 asserts that $r(T;k)>k(n-1)+1$ for every tree $T$ with $n\ge 1$ edges and $k$ sufficiently large such that $n$ divides $k-1$. Using a folklore double counting argument in set system and the edge chromatic number of complete graphs, we prove that if $k$ is odd and $n$ is sufficiently large compared with $m$ and $k$, then \[ r(S(n,m);k)=kn+m+2.\] This is a step in our effort to determine whether $r(S(n,m);k)$ and $r_\ell(S(n,m);k)$ are always equal, which remains wide open. We also prove that $ r(S^m_n;k)=k(n-1)+m+2$ if $k $ is odd and $n$ is sufficiently large compared with $m$ and $k$, where $1\le m\le n$ and $S^m_n$ is obtained from $K_{1, n}$ by subdividing $m$ edges each exactly once. We end the paper with some observations towards the list Ramsey number for $S(n,m)$ and $S^m_n$.