Homogeneous substructures in random ordered uniform matchings

TL;DR

This paper investigates the size of largest homogeneous substructures in random ordered uniform matchings, providing order-of-magnitude results for various pattern sets, including all small sets and specific r-partite patterns.

Contribution

It determines the order of magnitude of the largest $ ext{P}$-cliques in random ordered $r$-uniform matchings for several pattern sets, including all sets of size up to 2 and all r-partite patterns.

Findings

Largest $ ext{P}$-cliques have known order of magnitude for small pattern sets.

Results include all sets of size up to 2 and the set of all r-partite r-patterns.

Provides bounds and asymptotic behavior for these homogeneous substructures.

Abstract

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 $\tfrac12\binom{2r}r$ 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 $|\mathcal{P}|\le2$ and the set $\mathcal{R}^{(r)}$ of all $2^{r-1}$ $r$-partite $r$-patterns.