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arXiv 1802.08472math.CO
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随机图中的随机三角形

Random triangles in random graphs

Annika Heckel

备注：
4 pages; minor corrections, and updated references to new version of Oliver Riordan's paper
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摘要：

在最近的一篇论文中，Oliver Riordan表明，对于$r \ge 4$和$p$，只要达到并略大于$K_r$因子的阈值，由$G(n,p)$中的$K_r$的副本形成的超图包含具有$π\sim p^{r \choose 2}$的二项式随机超图$H=H_r(n,π)$的副本。对于$r=3$，他给出了一个稍弱的结果，其中密度在随机超图减少了一个常数因子。最近，Jeff Kahn提出了一个关于所有$r \ge 3$的Shamir超图匹配问题的阈值的渐近界。根据Riordan的结果，这立即意味着$r \ge 4$的$G(n,p)$中$K_r$ -因子的阈值的渐近锐界。在本文中，我们通过修改Riordan的论证来解决缺失的情况$r=3$。这意味着Kahn的结果也暗示了$G(n,p)$中三角形因子的一个严格界。

英文摘要：

In a recent paper, Oliver Riordan shows that for $r \ge 4$ and $p$ up to and slightly larger than the threshold for a $K_r$-factor, the hypergraph formed by the copies of $K_r$ in $G(n,p)$ contains a copy of the binomial random hypergraph $H=H_r(n,π)$ with $π\sim p^{r \choose 2}$. For $r=3$, he gives a slightly weaker result where the density in the random hypergraph is reduced by a constant factor. Recently, Jeff Kahn announced an asymptotically sharp bound for the threshold in Shamir's hypergraph matching problem for all $r \ge 3$. With Riordan's result, this immediately implies an asymptotically sharp bound for the threshold of a $K_r$-factor in $G(n,p)$ for $r \ge 4$. In this note, we resolve the missing case $r=3$ by modifying Riordan's argument. This means that Kahn's result also implies a sharp bound for triangle factors in $G(n,p)$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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