Multicolor Ramsey number for double stars

Abstract

For a graph H and an integer k≥1, let r(H;k) and rℓ(H;k) denote the k-color Ramsey number and list Ramsey number of H, respectively. Alon, Bucić, Kalvari, Kuperwasser and Szabó in 2021 initiated the systematic study of list Ramsey numbers of graphs and hypergraphs, and conjectured that r(K1,n;k) and rℓ(K1,n;k) are always equal. Motivated by their work, we study the k-color Ramsey number for double stars S(n,m), where n≥m≥1. To the best of our knowledge, little is known on the exact value of r(S(n,m);k) when k≥3. A classic result of Erdős and Graham from 1975 asserts that r(T;k)>k(n−1)+1 for every tree T with n≥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, thenr(S(n,m);k)=kn+m+2. This is a step in our effort to determine whether r(S(n,m);k) and rℓ(S(n,m);k) are always equal, which remains wide open. We also prove that r(Snm;k)=k(n−1)+m+2 if k is odd and n is sufficiently large compared with m and k, where 1≤m≤n and Snm is obtained from K1,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 Snm.