Homogeneous substructures in random ordered hyper-matchings

TL;DR

This paper investigates the maximum size of structured submatchings within random ordered hyper-matchings, providing bounds for various pattern sets and revealing the influence of pattern structure on submatching size.

Contribution

It characterizes the size of largest pattern-structured submatchings in random hyper-matchings for several pattern classes, including small sets, all $r$-partite patterns, and symmetric patterns.

Findings

Determined the size bounds for pattern sets of size up to 2.

Analyzed the maximum size of $ ext{P}$-cliques for all $r$-partite patterns.

Extended results to symmetric, Boolean-like pattern sets.

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 order-isomorphic to a member of $\mathcal{P}$. In this paper we are interested in the size of a largest $\mathcal{P}$-clique in a random ordered $r$-uniform matching selected uniformly from all such matchings on a fixed vertex set $[rn]$. We determine this size (up to multiplicative constants) for several sets $\mathcal{P}$, including all sets of size $|\mathcal{P}|\le2$, the set $\mathcal{R}^{(r)}$ of all $r$-partite patterns, as well as sets $\mathcal{P}$ enjoying a Boolean-like, symmetric structure.