随机超图中的分数团分解
Fractional clique decompositions in random hypergraphs
AI总结：
本文证明了随机超图在临界概率下以高概率允许分数团分解，改进了已有界，并推广到一般均匀超图，达到最优性。
AI中文摘要：
我们证明，当 $ p \ge n^{-1/2 + o(1)} $ 时，$ G(n, p) $ 以高概率允许分数三角形分解，即在其三角形上存在一个非负权重函数，使得包含每条边的所有三角形的总权重等于 1。这个关于 $ p $ 的界在渐近误差项内是最优的，改进了 Mahabaduge 和 Simkin 最近的最佳结果，即 $ p \ge n^{-4/11 + o(1)} $ 就足够了。我们的主要工具是一个确定性定理，该定理保证在所有满足适当“团正则性”性质的高阶图中存在分数团分解。我们通过分析 Mahabaduge 和 Simkin 提出的算法的一个扩展（并推广到高阶图）来证明这一点，在该算法中，在每个时间步，每条边的差异在其包含的三角形之间传播。通过展示随机 $ k $-均匀高阶图中相关量的集中性，我们得到对于所有 $ k \ge 2 $ 和 $ r \ge k + 1 $，当 $ p \ge n^{-\frac{r - k}{\binom{r}{k} - 1} + o(1)} $ 时，$ G^{(k)}(n, p) $ 以高概率允许分数 $ K^{(k)}_r $-分解，这改进了 Delcourt、Kelly 和 Postle 的结果，并且在子多项式因子内是最优的。
英文摘要：
We prove that, whenever $ p \ge n^{-1/2 + o(1)} $, with high probability $ G(n, p) $ admits a fractional triangle decomposition, that is, a non-negative weight function on its triangles for which the total weight of all triangles containing each edge is equal to 1. This bound on $ p $ is optimal up to the asymptotic error term, improving upon the recent state of the art, due to Mahabaduge and Simkin, that $ p \ge n^{-4/11 + o(1)} $ suffices. Our main tool is a deterministic theorem guaranteeing the existence of fractional clique decompositions in all hypergraphs satisfying suitable `clique-regularity' properties. We prove this by analysing an extension (and generalisation to hypergraphs) of an algorithm proposed by Mahabaduge and Simkin, in which, at each time step, the discrepancy at each edge is spread among its containing triangles. By showing the concentration of the relevant quantities in random $ k $-uniform hypergraphs, we obtain for all $ k \ge 2 $ and $ r \ge k + 1 $ that w.h.p. $ G^{(k)}(n, p) $ admits a fractional $ K^{(k)}_r $-decomposition whenever $ p \ge n^{-\frac{r - k}{\binom{r}{k} - 1} + o(1)} $, which improves upon results of Delcourt, Kelly, and Postle, and is best possible up to subpolynomial factors.
