We consider random simple temporal graphs in which every edge of the complete graph $K_n$ appears once within the time interval [0,1] independently and uniformly at random. Our main result is a sharp threshold on the size of any maximum $\delta$-clique (namely a clique with edges appearing at most $\delta$ apart within [0,1]) in random instances of this model, for any constant~$\delta$. In particular, using the probabilistic method, we prove that the size of a maximum $\delta$-clique is approximately $\frac{2\log{n}}{\log{\frac{1}{\delta}}}$ with high probability (whp). What seems surprising is that, even though the random simple temporal graph contains $\Theta(n^2)$ overlapping $\delta$-windows, which (when viewed separately) correspond to different random instances of the Erdos-Renyi random graphs model, the size of the maximum $\delta$-clique in the former model and the maximum clique size of the latter are approximately the same. Furthermore, we show that the minimum interval containing a $\delta$-clique is $\delta-o(\delta)$ whp. We use this result to show that any polynomial time algorithm for $\delta$-TEMPORAL CLIQUE is unlikely to have very large probability of success.

翻译：我们考虑一类随机简单时间图，其中完全图$K_n$的每条边都在时间区间[0,1]内独立且均匀随机地出现一次。我们的主要结果是：对于任意常数$\delta$，该模型随机实例中任意最大$\delta$-团（即边在[0,1]内出现时间间隔不超过$\delta$的团）的大小存在一个尖锐阈值。具体而言，我们采用概率方法证明：最大$\delta$-团的大小以高概率约为$\frac{2\log{n}}{\log{\frac{1}{\delta}}}$。令人惊讶的是，尽管随机简单时间图包含$\Theta(n^2)$个重叠的$\delta$-时间窗口（每个窗口单独视为Erdos-Renyi随机图模型的不同随机实例），但前者模型中最大$\delta$-团的大小与后者中的最大团大小近似相等。此外，我们证明包含一个$\delta$-团的最小时间区间以高概率为$\delta-o(\delta)$。该结果进一步表明：任何用于求解$\delta$-时间团问题的多项式时间算法都难以获得极高的成功概率。