An exponential improvement for Ramsey lower bounds

TL;DR

This paper establishes a new exponential lower bound on Ramsey numbers r(ℓ, Cℓ) for large ℓ, significantly improving previous bounds and revealing deeper combinatorial properties.

Contribution

It introduces the first exponential improvement over Erdős's classical lower bound for these Ramsey numbers, using a novel probabilistic approach.

Findings

New lower bound r(ℓ, Cℓ) ≥ (p_C^{-1/2} + ε)^ℓ for large ℓ

Identifies the unique solution p_C to C = (log p_C)/(log(1 - p_C))

Provides the first exponential improvement over Erdős's 1947 bound

Abstract

We prove a new lower bound on the Ramsey number $r(\ell, C\ell)$ for any constant $C > 1$ and sufficiently large $\ell$, showing that there exists $\varepsilon=\varepsilon(C)> 0$ such that \[ r(\ell, C\ell) \geq \left(p_C^{-1/2} + \varepsilon\right)^\ell, \] where $p_C \in (0, 1/2)$ is the unique solution to $C = \frac{\log p_C}{\log(1 - p_C)}$. This provides the first exponential improvement over the classical lower bound obtained by Erd\H{o}s in 1947.