Twins in ordered hyper-matchings

TL;DR

This paper investigates the size of largest twin sub-matchings in ordered hyper-matchings, establishing bounds and showing that for most such matchings, the maximum twin size aligns with the upper bound.

Contribution

The paper introduces new bounds on twin sizes in ordered hyper-matchings and demonstrates that typical matchings reach the upper bound, advancing understanding of their structure.

Findings

Lower bound of twins size: 6F5(n^{3/(5(2^{r-1}-1))})

Upper bound of twins size: 6F0(n^{2/(r+1)})

Most ordered r-matchings achieve the upper bound

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.