Packing nearly optimal Ramsey R(3,t) graphs
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In 1995 Kim famously proved the Ramsey bound R(3,t) > c t^2/ t by constructing an n-vertex graph that is triangle-free and has independence number at most C √(n 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 ϵ>0 we find an edge-disjoint collection (G_i)_i of n-vertex graphs G_i ⊆ K_n such that (a) each G_i is triangle-free and has independence number at most C_ϵ√(n n), and (b) the union of all the G_i contains at least (1-ϵ)n2 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) = Θ(r^2 r).
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