On the Ramsey multiplicity of complete graphs
David Conlon

TL;DR
This paper establishes a new lower bound on the number of monochromatic complete subgraphs in two-colored complete graphs, improving upon previous bounds by explicitly defining a smaller constant in the asymptotic estimate.
Contribution
The authors derive an improved asymptotic lower bound for the Ramsey multiplicity of complete graphs, with an explicitly calculated constant less than 4.
Findings
New lower bound with constant C ≈ 2.18
Improved asymptotic estimate over Erdős's bound
Explicit constant enhances understanding of monochromatic subgraph frequency
Abstract
We show that, for large, there must exist at least \[\frac{n^t}{C^{(1+o(1))t^2}}\] monochromatic s in any two-colouring of the edges of , where is an explicitly defined constant. The old lower bound, due to Erd\H{o}s \cite{E62}, and based upon the standard bounds for Ramsey's theorem, is \[\frac{n^t}{4^{(1+o(1))t^2}}.\]
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Topology and Set Theory · Advanced Graph Theory Research
