Optimal thresholds for Latin squares, Steiner Triple Systems, and edge colorings

TL;DR

This paper determines the threshold probability for the appearance of Latin squares, Steiner triple systems, and proper list edge-colorings in random hypergraphs and graphs, advancing understanding of their probabilistic existence.

Contribution

It establishes the precise threshold for Latin squares in random hypergraphs and extends similar results to Steiner triple systems and edge-colorings, answering several open questions.

Findings

Threshold for Latin squares is (log n/n)

Analogous thresholds for Steiner triple systems and edge-colorings

Addresses multiple open problems in combinatorics

Abstract

We show that the threshold for the binomial random $3$-partite, $3$-uniform hypergraph $G^{3}((n,n,n),p)$ to contain a Latin square is $\Theta(\log{n}/n)$. We also prove analogous results for Steiner triple systems and proper list edge-colorings of the complete (bipartite) graph with random lists. Our results answer several related questions of Johansson, Luria-Simkin, Casselgren-H\"aggkvist, Simkin, and Kang-Kelly-K\"uhn-Methuku-Osthus.