A lower bound on the Ramsey number $R_k(k+1,k+1)$

TL;DR

This paper establishes new lower bounds on certain multicolor Ramsey numbers using tower functions, significantly advancing the understanding of their growth rates.

Contribution

It provides novel lower bounds for specific Ramsey numbers involving multiple colors, employing tower functions to quantify their magnitude.

Findings

Proves that $R_k(k+1,k+1)$ is at least $4$ times a tower function of height roughly $k/4$.

Establishes similar lower bounds for $R_k(k+1,k+2)$, $R_k(k+1,2k+1)$, and $R_k(k+2,k+2)$.

Demonstrates the rapid growth of these Ramsey numbers with respect to $k$.

Abstract

We will prove that $R_k(k+1,k+1)\geq 4 tw_{\lfloor k/4\rfloor -3}(2)$, where $tw$ is the tower function defined by ${tw}_1(x)=x$ and ${tw}_{i+1}(x)=2^{{tw}_i(x)}$. We also give proofs of $R_k(k+1,k+2)\geq 4 tw_{k-7}(2)$, $R_k(k+1,2k+1)\geq 4 tw_{k-3}(2)$, and $R_k(k+2,k+2)\geq 4 tw_{k-4}(2)$.