Hypergraph coloring up to condensation
Peter Ayre, Amin Coja-Oghlan, Catherine Greenhill

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 -colorability threshold in random -uniform hypergraphs up to an additive error of , where . The new lower bound on the threshold matches the "condensation phase transition" predicted by statistical physics considerations [Krzakala et al., PNAS 2007].
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