On the existence of $\delta$-temporal cliques in random simple temporal graphs
<|reference_start|>On the existence of $\delta$-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 $\delta$-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.<|reference_end|>
@article{mertzios2024on,
    title={On the existence of $\delta$-temporal cliques in random simple temporal
  graphs},
    author={George B. Mertzios, Sotiris Nikoletseas, Christoforos Raptopoulos,
  Paul G. Spirakis},
    journal={arXiv preprint arXiv:2404.07147},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.07147},
    primaryClass={math.CO cs.DM}
}