Homogeneous substructures in random ordered hyper-matchings
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912883587153920 |
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| author | Dudek, Andrzej Grytczuk, Jarosław Przybyło, Jakub Ruciński, Andrzej |
| author_facet | Dudek, Andrzej Grytczuk, Jarosław Przybyło, Jakub Ruciński, Andrzej |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_20374 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Homogeneous substructures in random ordered hyper-matchings Dudek, Andrzej Grytczuk, Jarosław Przybyło, Jakub Ruciński, Andrzej Combinatorics 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. |
| title | Homogeneous substructures in random ordered hyper-matchings |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2507.20374 |