On the construction of non-2-colorable uniform hypergraphs

TL;DR

This paper investigates the minimal size of non-2-colorable uniform hypergraphs, providing new upper bounds for small n, including the first improvement for n=8, advancing understanding of hypergraph coloring limitations.

Contribution

The paper introduces new constructions that improve upper bounds on the minimal number of hyperedges in non-2-colorable hypergraphs for certain small values of n.

Findings

Improved upper bounds for m(n) for specific small n.

First known construction reducing the number of hyperedges for n=8.

Enhanced understanding of non-2-colorability in uniform hypergraphs.

Abstract

The problem of 2-coloring uniform hypergraphs has been extensively studied over the last few decades. An n-uniform hypergraph is not 2-colorable if its vertices can't be colored with two colors, Red and Blue, such that every hyperedge contains Red as well as Blue vertices. The least possible number of hyperedges in an n-uniform hypergraph which is not 2-colorable is denoted by m(n). In this paper, we consider the problem of finding an upper bound on m(n) for small values of n. We provide constructions which improve the existing results for some such values of n. We obtain the first improvement in the case of n=8.