On the existence of δ-temporal cliques in random simple temporal graphs

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 δ-clique (namely a clique with edges appearing at most δ apart within [0,1]) in random instances of this model, for any constant δ. In particular, using the probabilistic method, we prove that the size of a maximum δ-clique is approximately 2logn/log1/δ with high probability (whp). What seems surprising is that, even though the random simple temporal graph contains Θ(n^2) overlapping δ-windows, which (when viewed separately) correspond to different random instances of the Erdos-Renyi random graphs model, the size of the maximum δ-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 δ-clique is δ-o(δ) whp. We use this result to show that any polynomial time algorithm for δ-TEMPORAL CLIQUE is unlikely to have very large probability of success.