Packing Nearly Optimal Ramsey R(3,t) Graphs

Abstract

In 1995 Kim famously proved the Ramsey bound R(3, t) ≤ ct2/logt 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 ε > 0 we find an edge-disjoint collection (Gi)i of n-vertex graphs GiKn 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 \((1-\epsilon)\left(\begin{array}{l}n \\2\end{array}\right)\) 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, and 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(r2 logr).

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Correspondence to He Guo or Lutz Warnke.

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Research partially supported by NSF Grant DMS-1703516 and a Sloan Research Fellowship.

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Guo, H., Warnke, L. Packing Nearly Optimal Ramsey R(3,t) Graphs. Combinatorica 40, 63–103 (2020). https://doi.org/10.1007/s00493-019-3921-7

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Mathematics Subject Classification (2010)

  • 05C55
  • 05C80
  • 05D10
  • 60C05