Download the PDF of the paper by Dong Yeap Kang, Tom Kelly, Daniela K\”uhn, Abhishek Methuku, and Deryk Osthus titled Thresholds for Latin Squares and Steiner’s Triple Systems: Boundaries in Logarithmic Factors.

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overview: For $n \in \mathbb N$ and absolute constant $C$ , for $p \geq C\log^2 n / n$ and $L_{i,j} \subseteq , [n]$ is a random subset of $[n]$ where each $k\in [n]$ is contained separately in $L_{i,j}$ and with probability $p$ for each $i, j\in [n]For $, asymptotically almost certainly there exists a Latin square of order $n$ whose $i$th row and $j$th column entry is in $L_{i,j}$. The problem of determining the threshold probability that a Latin square of order $n exists was posed separately by Johansson, Luria and Simkin, and Casselgren and Hegqvist. Our results provide tight upper bounds up to a factor of $\log n$, strengthening the upper bound recently obtained by Sah, Sawhney, and Simkin. We also prove that similar results can be obtained for $1$ factorizations of Steiner triple systems and complete graphs, and further that each of these thresholds yields $1$ factorizations of nearly complete holomorphic bipartite graphs. is just a threshold for the existence of

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From: Thomas Kelly [view email]


Wed, Jun 29, 2022 09:01:13 UTC (57 KB)

Sun, Mar 26, 2023 21:09:01 UTC (127 KB)

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