On the multicolor Ramsey numbers of balanced double stars

The balanced double star on 2n+2 vertices, denoted S₍, ₍, 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 Rᵣ^bip (G) be the smallest integer N such that in every r-coloring of the edges of K₍, ₍ there is a monochromatic copy of G. It is known that R₂ (S₍, ₍) =3n+2 and R₂^bip (S₍, ₍) =2n+1 HJ, but very little is known about Rᵣ (S₍, ₍) and R^bipᵣ (S₍, ₍) 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 Rᵣ (S₍, ₍) (r-12) (2n+2) -1, \ (2r-4) n+1 R^bipᵣ (S₍, ₍) (2r-3+2r+O (1r²) ) n. \ These bounds are similar to the best known bounds on Rᵣ (P₂₍+₂) and Rᵣ^bip (P₂₍+₂), where P₂₍+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 R^bipᵣ (S₍, ₍) when r=3 and r=5.