On the size of temporal cliques in subcritical random temporal graphs

A random temporal graph is an Erdos-R\'enyi random graph G (n, p), together with a random ordering of its edges. A path in the graph is called increasing if the edges on the path appear in increasing order. A set S of vertices forms a temporal clique if for all u, v S, there is an increasing path from u to v. Becker2023 proved that if p=c n/n for c>1, then, with high probability, there is a temporal clique of size n-o (n). On the other hand, for c<1, with high probability, the largest temporal clique is of size o (n). In this note we improve the latter bound by showing that, for c<1, the largest temporal clique is of constant size with high probability.