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arXiv · 2206.14472

Thresholds for Latin squares and Steiner triple systems: Bounds within a logarithmic factor

Dong Yeap Kang·Tom Kelly·Daniela Kühn·Abhishek Methuku·Deryk Osthus
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Abstract

We prove that for $n \in \mathbb N$ and an absolute constant $C$, if $p \geq C\log^2 n / n$ and $L_{i,j} \subseteq [n]$ is a random subset of $[n]$ where each $k\in [n]$ is included in $L_{i,j}$ independently with probability $p$ for each $i, j\in [n]$, then asymptotically almost surely there is an order-$n$ Latin square in which the entry in the $i$th row and $j$th column lies in $L_{i,j}$. The problem of determining the threshold probability for the existence of an order-$n$ Latin square was raised independently by Johansson, by Luria and Simkin, and by Casselgren and H{\"a}ggkvist; our result provides an upper bound which is tight up to a factor of $\log n$ and strengthens the bound recently obtained by Sah, Sawhney, and Simkin. We also prove analogous results for Steiner triple systems and $1$-factorizations of complete graphs, and moreover, we show that each of these thresholds is at most the threshold for the existence of a $1$-factorization of a nearly complete regular bipartite graph.

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Dong Yeap Kang, Tom Kelly, Daniela Kühn, Abhishek Methuku, Deryk Osthus. 2022-06-29. Thresholds for Latin squares and Steiner triple systems: Bounds within a logarithmic factor. https://arxiv.org/abs/2206.14472

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