An extremal problem in proper $(r,p)$-coloring of hypergraphs

TL;DR

This paper investigates the maximum number of hyperedges in a hypergraph that can be properly colored with an $r$-coloring such that each hyperedge contains at least $p$ distinct colors, exploring extremal structures.

Contribution

It introduces a new extremal problem in hypergraph coloring, analyzing the maximum count of properly $(r,p)$ colored hyperedges and characterizing optimal structures.

Findings

Determined the maximum number of hyperedges properly $(r,p)$ colored.

Characterized structures that achieve this maximum.

Provided bounds and conditions for extremal hypergraph configurations.

Abstract

Let $G(V,E)$ be a $k$-uniform hypergraph. A hyperedge $e \in E$ is said to be properly $(r,p)$ colored by an $r$-coloring of vertices in $V$ if $e$ contains vertices of at least $p$ distinct colors in the $r$-coloring. An $r$-coloring of vertices in $V$ is called a {\it strong $(r,p)$ coloring} if every hyperedge $e \in E$ is properly $(r,p)$ colored by the $r$-coloring. We study the maximum number of hyperedges that can be properly $(r,p)$ colored by a single $r$-coloring and the structures that maximizes number of properly $(r,p)$ colored hyperedges.