A note on multicolour Ramsey numbers and random sphere graphs

TL;DR

The paper addresses improving lower bounds for multicolour Ramsey numbers $r(t;\ell)$ with fixed $\ell\ge 3$. It replaces the binomial random graph in the Sawin–Wigderson framework with random sphere graphs $G_{k,p}(n)$, leveraging geometric concentration to strengthen the exponent in the lower bounds. The main result shows there exists $\varepsilon>0$ such that for every fixed $\ell\ge 3$, $r(t;\ell) \ge 2^{(\delta_*+\varepsilon)(\ell-2)t + t/2 - o(t)}$, with $\delta_* \approx 0.383796$. This advances the asymptotic landscape of Ramsey bounds and highlights the effectiveness of sphere-graph constructions in this combinatorial setting.

Abstract

The Ramsey number $r(t;\ell)$ is the smallest $n$ such that every $\ell$-coloring of the edges of $K_n$ gives a monochromatic $K_{t}$. In recent years, there have been several improvements on asymptotic lower bounds for these numbers when $\ell\geq 3$. This started with a breakthrough result of Conlon and Ferber, followed by further improvements of Wigderson and then Sawin. Building on the previous approaches, Sawin used blowups of an unbalanced binomial random graph to show that there is some explicit constant $δ_*\approx 0.383796$ such that $r(t;\ell)\geq 2^{δ_*(\ell-2)t+t/2+o(t)}$. In this short note, we show that one can get an exponential improvement in this bound by replacing the use of a binomial random graph with a random sphere graph, a model which which has recently been applied by Ma, Shen and Xie in a breakthrough on lower bounds for (2-colour) Ramsey numbers in the (slightly) off-diagonal setting.

Theorems & Definitions (10)

• Theorem 1
• Remark 1
• Definition 1
• Lemma 1: SAWIN, Lemma 1
• Lemma 2: SAWIN, Lemma 2
• Lemma 3: MASHENXIE, Lemma 4.1
• Definition 2
• Lemma 4: MASHENXIE, Theorem 7.1 and proof of Lemma 9.2
• Theorem 2
• proof