A new lower bound for the multicolor Ramsey number $r_k(K_{2, t + 1})$

TL;DR

This paper introduces a new family of $K_{2, t+1}$-free graphs and uses them to establish improved lower bounds for the multicolor Ramsey number $r_k(K_{2, t+1})$ when $k$ and $t$ are powers of the same prime.

Contribution

It provides a novel construction of $K_{2, t+1}$-free graphs and derives tighter lower bounds for the multicolor Ramsey number in specific cases.

Findings

Established a new infinite family of $K_{2, t+1}$-free graphs.

Derived lower bounds $tk^2 + 1$ for $r_k(K_{2, t+1})$ when $k$ and $t$ are prime powers.

Compared new bounds with existing upper bounds to improve understanding of $r_k(K_{2, t+1})$.

Abstract

In this short note, we provide a new infinite family of $K_{2, t+1}$-free graphs for each prime power $t$. Using these graphs, we show that it is possible to partition the edges of $K_n$ into parts, such that each part is isomorphic to our $K_{2, t+1}$-free graph. This yields an improved lower bound to the multicolor Ramsey number $r_k(K_{2, t+1})$ when $k$ and $t$ are powers of the same prime. For these values of $k$ and $t$, our coloring implies that $$ tk^2 + 1 \leq r_k(K_{2, t+1}) \leq tk^2 + k + 2. $$ where the upper bound is due to Chung and Graham.