Lower bounds for the chromatic number of certain Kneser-type hypergraphs

TL;DR

This paper establishes new lower bounds for the chromatic number of Kneser-type hypergraphs, extending existing results by incorporating intersection allowances among vertices and generalizing previous theorems.

Contribution

It introduces an extended equitable r-colorability defect for hypergraphs with intersecting vertices and provides generalized lower bounds for their chromatic number.

Findings

Derived lower bounds for chromatic number based on extended equitable defect.

Generalized previous results to broader families of subsets.

Connected intersection properties with hypergraph coloring complexity.

Abstract

Let $n\ge 1$, $r\ge 2$, and $s\ge 0$ be integers and ${\cal P}=\{P_1,\dots, P_l\}$ be a partition of $[n]=\{1,\dots, n\}$ with $|P_i|\le r$ for $i=1,\dots, l$. Also, let $\cal F$ be a family of non-empty subsets of $[n]$. The $r$-uniform Kneser-type hypergraph $\mbox{KG}^r({\cal F}, {\cal P},s)$ is the hypergraph with the vertex set of all $\cal P$-admissible elements $F\in {\cal F}$, that is $|F\cap P_i|\le 1$ for $i=1,\dots, l$ and the edge set of all $r$-subsets $\{F_1,\dots, F_r\}$ of the vertex set that $|F_i\cap F_j|\le s$ for all $1\le i<j\le r$. In this article, we extend the equitable $r$-colorability defect $\mbox{ecd}^r({\cal F})$ of Abyazi Sani and Alishahi to the case when one allows intersection among the vertices of an edge. It will be denoted by $\mbox{ecd}^r({\cal F},s)$. We then, give (under certain assumptions) lower bounds for the chromatic number of $\mbox{KG}^r({\cal F}, {\cal P},s)$ and some of its variants in terms of $\mbox{ecd}^r({\cal F},\lfloor s/2\rfloor)$. This work generalizes many existing results in the literature of the Kneser hypergraphs. It generalizes the previous results of the current authors from the special family of all $k$-subsets of $[n]$ to a general family $\cal F$ of subsets.