Packing nearly optimal Ramsey R(3,t) graphs

TL;DR

This paper extends Kim's 1995 construction of nearly optimal Ramsey R(3,t) graphs by approximately decomposing the complete graph into edge-disjoint, triangle-free graphs with bounded independence number, using a semi-random process.

Contribution

It introduces an algorithmic method to decompose K_n into nearly optimal Ramsey graphs, advancing understanding of graph packings and Ramsey parameters.

Findings

Constructed edge-disjoint triangle-free graphs covering almost all edges of K_n.

Proved the existence of such packings using a semi-random triangle-free process.

Resolved a conjecture relating to the minimum degree of Ramsey minimal graphs for triangles.

Abstract

In 1995 Kim famously proved the Ramsey bound R(3,t) \ge c t^2/\log t by constructing an n-vertex graph that is triangle-free and has independence number at most C \sqrt{n \log n}. We extend this celebrated result, which is best possible up to the value of the constants, by approximately decomposing the complete graph K_n into a packing of such nearly optimal Ramsey R(3,t) graphs. More precisely, for any \epsilon>0 we find an edge-disjoint collection (G_i)_i of n-vertex graphs G_i \subseteq K_n such that (a) each G_i is triangle-free and has independence number at most C_\epsilon \sqrt{n \log n}, and (b) the union of all the G_i contains at least (1-\epsilon)\binom{n}{2} edges. Our algorithmic proof proceeds by sequentially choosing the graphs G_i via a semi-random (i.e., Rodl nibble type) variation of the triangle-free process. As an application, we prove a conjecture in Ramsey theory by Fox, Grinshpun, Liebenau, Person, and Szabo (concerning a Ramsey-type parameter introduced by Burr, Erdos, Lovasz in 1976). Namely, denoting by s_r(H) the smallest minimum degree of r-Ramsey minimal graphs for H, we close the existing logarithmic gap for H=K_3 and establish that s_r(K_3) = \Theta(r^2 \log r).