Erdős-Szekeres type theorems for ordered uniform matchings

For $r,n⩾2$, an ordered r-uniform matching of size n is an r-uniform hypergraph on a linearly ordered vertex set V, with $\left|V\right|=rn$, consisting of n pairwise disjoint edges. There are $\frac{1}{2}\left(\begin{array}{c}2r\\ r\end{array}\right)$ different ways two edges may intertwine, called here patterns. Among them we identify $3^{r-1}$ collectable patterns P, which have the potential of appearing in arbitrarily large quantities called P-cliques.

We prove an Erdős-Szekeres type result guaranteeing in every ordered r-uniform matching the presence of a P-clique of a prescribed size, for some collectable pattern P. In particular, in the diagonal case, one of the P-cliques must be of size $\Omega \left(n^{3^{1-r}}\right)$. In addition, for each collectable pattern P we show that the largest size of a P-clique in a random ordered r-uniform matching of size n is, with high probability, $\Theta \left(n^{1/r}\right)$.