The (p,q)-extremal problem and the fractional chromatic number of Kneser hypergraphs

TL;DR

This paper establishes a connection between the fractional chromatic number of Kneser hypergraphs and the $(p,q)$-extremal problem, providing solutions for large graphs and calculating the fractional chromatic number for specific cases.

Contribution

It links the fractional chromatic number of Kneser hypergraphs to the $(p,q)$-extremal problem and solves this problem for large graphs with certain parameters.

Findings

Solved the $(p,q)$-extremal problem for large graphs with $p \,\geq\, q \,\geq\, 3$

Calculated the fractional chromatic number of Kneser hypergraphs with sets of size 2

Established a new connection between extremal hypergraph problems and chromatic numbers.

Abstract

The problem of computing the chromatic number of Kneser hypergraphs has been extensively studied over the last 40 years and the fractional version of the chromatic number of Kneser hypergraphs is only solved for particular cases. The \emph{$(p,q)$-extremal problem} consists in finding the maximum number of edges on a $k$-uniform hypergraph $\mathcal{H}$ with $n$ vertices such that among any $p$ edges some $q$ of them have no empty intersection. In this paper we have found a link between the fractional chromatic number of Kneser hypergraphs and the $(p,q)$-extremal problem and also solve the $(p,q)$-extremal problem for graphs if $n$ is sufficiently large and $p \geq q \geq 3$ by proposing it as a problem of extremal graph theory. With the aid of this result we calculate the fractional chromatic number of Kneser hypergraphs when they are composed with sets of cardinality 2.