Hypergraphs with many Kneser colorings (Extended Version)

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 Erd
53s-Rothschild problems.

Abstract

For fixed positive integers $r, k$ and $\ell$ with $1 \leq \ell < r$ and an $r$-uniform hypergraph $H$, let $\kappa (H, k,\ell)$ denote the number of $k$-colorings of the set of hyperedges of $H$ for which any two hyperedges in the same color class intersect in at least $\ell$ elements. Consider the function $\KC(n,r,k,\ell)=\max_{H\in{\mathcal H}_{n}} \kappa (H, k,\ell) $, where the maximum runs over the family ${\mathcal H}_n$ of all $r$-uniform hypergraphs on $n$ vertices. In this paper, we determine the asymptotic behavior of the function $\KC(n,r,k,\ell)$ for every fixed $r$, $k$ and $\ell$ 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, Quarterly Journal of Mathematics, Oxford Series, Series 2, {\bf 12} (1961), 313--320].