range in logarithmic factor

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