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arXiv · 2507.20374

Homogeneous substructures in random ordered hyper-matchings

Andrzej Dudek·Jarosław Grytczuk·Jakub Przybyło·Andrzej Ruciński
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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.

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Andrzej Dudek, Jarosław Grytczuk, Jakub Przybyło, Andrzej Ruciński. 2025-07-27. Homogeneous substructures in random ordered hyper-matchings. https://arxiv.org/abs/2507.20374

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