Random hypergraphs and property B

TL;DR

This paper investigates the threshold for two-colorability in random hypergraphs, establishing that below a certain edge density, hypergraphs are almost surely two-colorable, and presents an efficient coloring algorithm for dense hypergraphs.

Contribution

It proves that hypergraphs with slightly fewer than Erdős' bound are almost surely two-colorable and introduces a simple greedy algorithm effective near the theoretical threshold.

Findings

Hypergraphs with fewer than (1 - ε) times Erdős' bound are almost surely two-colorable.

A greedy coloring algorithm successfully colors hypergraphs with up to O(k ln k 2^k) edges.

The results extend understanding of phase transitions in random hypergraph colorability.

Abstract

In 1964 Erd\H{o}s proved that $(1+\oh{1})) \frac{\eul \ln(2)}{4} k^2 2^{k}$ edges are sufficient to build a $k$-graph which is not two colorable. To this day, it is not known whether there exist such $k$-graphs with smaller number of edges. Erd\H{o}s' bound is consequence of the fact that a hypergraph with $k^2/2$ vertices and $M(k)=(1+\oh{1}) \frac{\eul \ln(2)}{4} k^2 2^{k}$ randomly chosen edges of size $k$ is asymptotically almost surely not two colorable. Our first main result implies that for any $\varepsilon > 0$, any $k$-graph with $(1-\varepsilon) M(k)$ randomly and uniformly chosen edges is a.a.s. two colorable. The presented proof is an adaptation of the second moment method analogous to the developments of Achlioptas and Moore from 2002 who considered the problem with fixed size of edges and number of vertices tending to infinity. In the second part of the paper we consider the problem of algorithmic coloring of random $k$-graphs. We show that quite simple, and somewhat greedy procedure, a.a.s. finds a proper two coloring for random $k$-graphs on $k^2/2$ vertices, with at most $\Oh{k\ln k\cdot 2^k}$ edges. That is of the same asymptotic order as the analogue of the \emph{algorithmic barrier} defined by Achlioptas and Coja-Oghlan in 2008, for the case of fixed $k$.