Homogeneous substructures in random ordered uniform matchings

An ordered $r$-uniform matching of size $n$ is a collection of $n$ pairwise disjoint $r$-subsets of a linearly ordered set of $rn$ vertices. For $n=2$, such a matching is called an $r$-pattern, as it represents one of $\frac{1}{2}\left(\genfrac{}{}{0px}{}{2r}{r}\right)$ ways two disjoint edges may intertwine. Given a set $\mathcal{P}$ of $r$-patterns, a $\mathcal{P}$-clique is a matching with all pairs of edges belonging to $\mathcal{P}$. In this paper we determine the order of magnitude of the size of a largest $\mathcal{P}$-clique in a random ordered $r$-uniform matching for several sets $\mathcal{P}$, including all sets of size $\mathrm{\mid }\mathcal{P}\mathrm{\mid }\le 2$ and the set ${\mathcal{R}}^{(r)}$ of all $2^{r-1}$ $r$-partite $r$-patterns.