On some generalizations of the property $B$-problem of an $n$-uniform hypergraph

TL;DR

This paper investigates extremal hypergraph coloring problems related to the Erdős–Hajnal property, providing new lower bounds for the minimal number of edges in certain uniform hypergraphs that resist specific 2-colorings.

Contribution

It introduces new lower bounds for the minimal edge count in hypergraphs that do not admit certain 2-colorings, generalizing the property B-problem.

Findings

Derived new lower bounds for $m_k(n)$.

Extended understanding of hypergraph coloring constraints.

Contributed to extremal combinatorics theory.

Abstract

The extremal problem of hypergraph colorings related to Erd\H{o}s--Hajnal property $B$-problem is considered. Let $k$ be a natural number. The problem is to find the value of $m_k(n)$ equal to the minimal number of edges in an $n$-uniform hypergraph not admitting $2$-colorings of the vertex set such that every edge of the hypergraph contains at least $k$ vertices of each color. In this paper we obtain new lower bounds for $m_k(n).$