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The Four-Color Ramsey Multiplicity of Triangles

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arxiv 2312.08049 v1 pith:H4TO4Q6M submitted 2023-12-13 math.CO math.OC

Aldo Kiem , Sebastian Pokutta , Christoph Spiegel This is my paper · ORCID
classification math.COmath.OC
keywords resulttrianglesalgebraallowsasymptoticallyblow-upcombinatorialcomplete
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abstract

We study a generalization of a famous result of Goodman and establish that asymptotically at least a $1/256$ fraction of all triangles needs to be monochromatic in any four-coloring of the edges of a complete graph. We also show that any large enough extremal construction must be based on a blow-up of one of the two $R(3,3,3)$ Ramsey-colorings of $K_{16}$. This result is obtained through an efficient flag algebra formulation by exploiting problem-specific combinatorial symmetries that also allows us to study some related problems.

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  1. Density of rainbow triangles and properly colored $K_4$'s

    math.CO 2025-11 conditional novelty 6.0 of 10

    A graph with R red, G green, B blue edges contains at most ¼(RGB)^{2/3} properly colored K4s, with equality only for balanced blowups of a properly colored K4; the known rainbow-triangle bound √(2RGB) receives new fla...

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