On the multicolor Ramsey numbers of balanced double stars

Abstract

The balanced double star on $2n+2$ vertices, denoted $S_{n,n}$, is the tree obtained by joining the centers of two disjoint stars each having $n$ leaves. Let $R_r(G)$ be the smallest integer $N$ such that in every $r$-coloring of the edges of $K_N$ there is a monochromatic copy of $G$, and let $R_r^{\mathrm{bip}}(G)$ be the smallest integer $N$ such that in every $r$-coloring of the edges of $K_{N,N}$ there is a monochromatic copy of $G$. It is known that $R_2(S_{n,n})=3n+2$ and $R_2^{\mathrm{bip}}(S_{n,n})=2n+1$ \cite{HJ}, but very little is known about $R_r(S_{n,n})$ and $R^{\mathrm{bip}}_r(S_{n,n})$ when $r\geq 3$ (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 $n\geq 1$ (where the lower bounds are adapted from existing examples): \[(r-1)2n+1\leq R_r(S_{n,n})\leq (r-\frac{1}{2})(2n+2)-1,\]and \[(2r-4)n+1\leq R^{\mathrm{bip}}_r(S_{n,n})\leq (2r-3+\frac{2}{r}+O(\frac{1}{r^2}))n.\] These bounds are similar to the best known bounds on $R_r(P_{2n+2})$ and $R_r^{\mathrm{bip}}(P_{2n+2})$, where $P_{2n+2}$ is a path on $2n+2$ vertices (which is also a balanced tree). We also give an example which improves the lower bound on $R^{\mathrm{bip}}_r(S_{n,n})$ when $r=3$ and $r=5$.

Figures (1)

• Figure 1: The edge coloring between sets used in the construction of the proof of Example \ref{['ex:small']} with three colors (left), four colors (center), and five colors (right). The size of each set is listed below the set name (ignoring floors and ceilings); for example, when $r=3$, $|X_1|= (1+\sqrt{2})n$.

Theorems & Definitions (27)

• Example 1.2
• proof
• Lemma 1.4
• proof : Proof of Observation \ref{['obs:ES']}
• Theorem 1.5
• Theorem 1.7: Decamillis and Song DS
• Corollary 1.8: Decamillis and Song DS
• Theorem 1.9
• Example 1.10
• Example 1.11