Hypergraphs with many Kneser colorings (Extended Version)
Carlos Hoppen, Yoshiharu Kohayakawa, Hanno Lefmann

TL;DR
This paper determines the asymptotic maximum number of k-colorings with intersecting hyperedges in r-uniform hypergraphs, extending classical intersection theorems and identifying extremal structures.
Contribution
It provides the asymptotic behavior of the maximum number of such colorings for fixed parameters and characterizes the extremal hypergraphs.
Findings
Asymptotic formula for (n,r,k,) for fixed r, k, .
Identification of extremal hypergraphs achieving the maximum.
Connection to classical intersection theorems and Erd53s-Rothschild problems.
Abstract
For fixed positive integers and with and an -uniform hypergraph , let denote the number of -colorings of the set of hyperedges of for which any two hyperedges in the same color class intersect in at least elements. Consider the function , where the maximum runs over the family of all -uniform hypergraphs on vertices. In this paper, we determine the asymptotic behavior of the function for every fixed , and and describe the extremal hypergraphs. This variant of a problem of Erd\H{o}s and Rothschild, who considered edge colorings of graphs without a monochromatic triangle, is related to the Erd\H{o}s--Ko--Rado Theorem on intersecting systems of sets [Intersection Theorems for Systems of Finite Sets,…
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Taxonomy
TopicsLimits and Structures in Graph Theory · Graph Labeling and Dimension Problems · Advanced Graph Theory Research
