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arXiv · 2310.01394

Twins in ordered hyper-matchings

Andrzej Dudek·Jarosław Grytczuk·Andrzej Ruciński
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Abstract

An ordered $r$-matching of size $n$ is an $r$-uniform hypergraph on a linearly ordered set of vertices, consisting of $n$ pairwise disjoint edges. Two ordered $r$-matchings are isomorphic if there is an order-preserving isomorphism between them. A pair of twins in an ordered $r$-matching is formed by two vertex disjoint isomorphic sub-matchings. Let $t^{(r)}(n)$ denote the maximum size of twins one may find in every ordered $r$-matching of size $n$. By relating the problem to that of largest twins in permutations and applying some recent Erd\H{o}s-Szekeres-type results for ordered matchings, we show that $t^{(r)}(n)=\Omega\left(n^{\frac{3}{5\cdot(2^{r-1}-1)}}\right)$ for every fixed $r\geqslant 2$. On the other hand, $t^{(r)}(n)=O\left(n^{\frac{2}{r+1}}\right)$, by a simple probabilistic argument. As our main result, we prove that, for almost all ordered $r$-matchings of size $n$, the size of the largest twins achieves this bound.

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Andrzej Dudek, Jarosław Grytczuk, Andrzej Ruciński. 2023-10-02. Twins in ordered hyper-matchings. https://arxiv.org/abs/2310.01394

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