Hypergraph coloring up to condensation

TL;DR

This paper refines the understanding of the q-colorability threshold in random k-uniform hypergraphs, aligning it with the condensation phase transition predicted by statistical physics, and improves previous bounds.

Contribution

It determines the q-colorability threshold up to a small additive error, matching the predicted condensation phase transition, advancing the theoretical understanding of hypergraph coloring.

Findings

Established the q-colorability threshold with high precision.

Matched the lower bound to the condensation phase transition.

Improved upon previous bounds by Dyer, Frieze, and Greenhill.

Abstract

Improving a result of Dyer, Frieze and Greenhill [Journal of Combinatorial Theory, Series B, 2015], we determine the $q$-colorability threshold in random $k$-uniform hypergraphs up to an additive error of $\ln 2+\varepsilon_q$, where $\lim_{q\to\infty}\varepsilon_q=0$. The new lower bound on the threshold matches the "condensation phase transition" predicted by statistical physics considerations [Krzakala et al., PNAS 2007].