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

Given a graph G, a random (k, n)-list assignment L for edges of G is an assignment of an independent, uniformly random set of colors to each edge e and a proper L-list coloring of G is a proper edge-coloring where the color of an edge e belongs to L(e). We show that for a random (O(log n), n)-list assignment L for edges of the complete bipartite graph Kn,n, there is a an L-list coloring of Kn,n with high probability. We also prove analogous results for the thresholds of Steiner triple systems and Latin squares in random (binomial) hypergraphs. All of our results are optimal up to absolute constants, and resolve several related conjectures of Johansson, Luria-Simkin, Casselgren-Häggkvist, Simkin, and Kang-Kelly-Kühn-Methuku-Osthus.