An update on multicolor Ramsey lower bounds

TL;DR

The paper addresses improving lower bounds for multicolor Ramsey numbers $r(t;\ell)$ by refining the upper bound on $c_{t,t}$, the probability that $t$ random vertices form an independent set in a $K_t$-free graph. It replaces the Erdős–Rényi subgraph used in Sawin's construction with the Ma--Shen--Xie spherical random geometric graph, exploiting correlations that suppress independent sets and modestly enhance cliques. The authors derive a bound $c_{t,t}\le\exp(-\alpha(p,D)t^2+o(t^2))$ with $\alpha(p,D)$ computable via a minimization over $\theta$ and show that, for $p\in(0.42,0.5)$ and large dimension parameter $D$, this bound yields a strict improvement over the Erdős–Rényi-based approach, translating into a small exponential improvement in $r(t;\ell)$ for all $\ell\ge3$. Consequently, the work provides new explicit lower bounds on off-diagonal multicolor Ramsey numbers in the high-$t$ regime, advancing understanding of Ramsey phenomena via high-dimensional geometric constructions.

Abstract

Building upon previous works by Conlon-Ferber and Wigderson, Sawin showed a few years ago that upper bounds on the minimum density of independent sets in a $K_t$-free $G$ can be used to provide lower bounds for multicolor Ramsey numbers. In this note, we observe how a further improved upper bound on this parameter directly follows from a recent spherical random geometric graph construction of Ma-Shen-Xie. As a consequence, we derive a small exponential improvement over the best known lower bounds for multicolor Ramsey numbers.

Theorems & Definitions (7)

• Theorem 1: Sawin
• Theorem 2
• Definition 3: Spherical random geometric graph $H(M,d,p)$
• Proposition 4
• Theorem 5
• proof
• proof : Proof of Theorem \ref{['cor:main-numerics']}