Improved Lower Bounds for Property B

TL;DR

This paper improves the lower bounds on the minimal size of hypergraphs lacking property B for small n, refining previous probabilistic methods and analyzing hypergraphs with a fixed number of vertices.

Contribution

It introduces a refined probabilistic approach to establish tighter lower bounds for small n and studies hypergraphs with a fixed number of vertices without property B.

Findings

Improved lower bounds for small n hypergraphs without property B.

Enhanced probabilistic methods based on refined random greedy coloring.

Analysis of hypergraphs with a fixed number of vertices lacking property B.

Abstract

If an $n$-uniform hypergraph can be 2-colored, then it is said to have property B. Erd\H{o}s (1963) was the first to give lower and upper bounds for the minimal size $m(n)$ of an $n$-uniform hypergraph without property B. His asymptotic upper bound $O(n^22^n)$ still is the best we know, his lower bound $2^{n-1}$ has seen a number of improvements, with the current best $\Omega$ $(2^n\sqrt{n/\log(n)})$ established by Radhakrishnan and Srinivasan (2000). Cherkashin and Kozik (2014) provided a simplified proof of this result, using Pluh\'ar's (2009) idea of a random greedy coloring. In the present paper, we use a refined version of this argument to obtain improved lower bounds on $m(n)$ for small values of $n$. We also study $m(n,v)$, the size of the smallest $n$-hypergraph without property B having $v$ vertices.