Packing nearly optimal Ramsey R(3,t) graphs

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 Kn into a packing of such nearly optimal Ramsey R(3,t) graphs. More precisely, for any \epsilon>0 we find an edge-disjoint collection (Gi)i of n-vertex graphs Gi \subseteq Kn such that (a) each Gi is triangle-free and has independence number at most C\epsilon \sqrt{n \log n}, and (b) the union of all the Gi contains at least (1-\epsilon)\binom{n}{2} edges. Our algorithmic proof proceeds by sequentially choosing the graphs Gi 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 sr(H) the smallest minimum degree of r-Ramsey minimal graphs for H, we close the existing logarithmic gap for H=K3 and establish that sr(K_3) = \Theta(r² \log r).