A note on two-colorability of nonuniform hypergraphs

TL;DR

This paper improves bounds on two-colorability of nonuniform hypergraphs by showing a random greedy coloring method likely finds a proper coloring when the expected monochromatic edges are bounded by a logarithmic function of the minimum edge size.

Contribution

It demonstrates that a simple random greedy coloring algorithm can efficiently find a proper two-coloring under weaker conditions than previously known, with bounds proportional to the logarithm of the minimum edge size.

Findings

Random greedy coloring likely finds a proper coloring under new bounds.

Improves previous results by reducing the bound from logarith-star to logarithmic.

Establishes a constant factor bound for two-colorability based on expected monochromatic edges.

Abstract

For a hypergraph $H$, let $q(H)$ denote the expected number of monochromatic edges when the color of each vertex in $H$ is sampled uniformly at random from the set of size 2. Let $s_{\min}(H)$ denote the minimum size of an edge in $H$. Erd\H{o}s asked in 1963 whether there exists an unbounded function $g(k)$ such that any hypergraph $H$ with $s_{\min}(H) \geq k$ and $q(H) \leq g(k)$ is two colorable. Beck in 1978 answered this question in the affirmative for a function $g(k) = \Theta(\log^* k)$. We improve this result by showing that, for an absolute constant $\delta>0$, a version of random greedy coloring procedure is likely to find a proper two coloring for any hypergraph $H$ with $s_{\min}(H) \geq k$ and $q(H) \leq \delta \cdot \log k$.