Packing nearly optimal Ramsey R(3, t) graphs

In 1995 Kim famously proved the Ramsey bound R(3, t) ≥ ct/ log t by constructing an n-vertex graph that is triangle-free and has independence number at most C √ 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 ǫ > 0 we find an edge-disjoint collection (Gi)i of n-vertex graphs Gi ⊆ Kn such that (a) each Gi is triangle-free and has independence number at most Cǫ √ n log n, and (b) the union of all the Gi contains at least (1− ǫ) ( n 2 ) edges. Our algorithmic proof proceeds by sequentially choosing the graphs Gi via a semi-random (i.e., Rödl nibble type) variation of the triangle-free process. As an application, we prove a conjecture in Ramsey theory by Fox, Grinshpun, Liebenau, Person, and Szabó (concerning a Ramsey-type parameter introduced by Burr, Erdős, Lovász 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(K3) = Θ(r 2 log r).