On the multicolor Ramsey numbers of balanced double stars

The balanced double star on $2n+2$ vertices, denoted Sn,n, is the tree obtained by joining the centers of two disjoint stars each having n leaves. Let Rᵣ(G) be the smallest integer N such that in every r-coloring of the edges of KN there is a monochromatic copy of G, and let Rᵣᵇⁱᵖ(G) be the smallest integer N such that in every r-coloring of the edges of KN,N there is a monochromatic copy of G. It is known that R₂(Sn,n)=3n+2 and R₂ᵇⁱᵖ(Sn,n)=2n+1 {HJ}, but very little is known about Rᵣ(Sn,n) and Rᵇⁱᵖᵣ(Sn,n) when r≥ 3 (other than the bounds which follow from considerations on the number of edges in the majority color class). In this paper we prove the following for all n≥ 1 (where the lower bounds are adapted from existing examples): \[(r-1)2n+1≤ R_r(Sn,n)≤ (r-1/2)(2n+2)-1,\]and \[(2r-4)n+1≤ Rᵇⁱᵖ_r(Sn,n)≤ (2r-3+2/r+O(1/r^2))n.\] These bounds are similar to the best known bounds on Rᵣ(P₂ₙ₊₂) and Rᵣᵇⁱᵖ(P₂ₙ₊₂), where P₂ₙ₊₂ is a path on $2n+2$ vertices (which is also a balanced tree). We also give an example which improves the lower bound on Rᵇⁱᵖᵣ(Sn,n) when $r=3$ and $r=5$.