Ordered unavoidable sub-structures in matchings and random matchings

TL;DR

This paper investigates unavoidable sub-structures in ordered matchings, establishing size bounds for certain configurations and analyzing their occurrence in random matchings, with extensions to uniform matchings and multiple twins.

Contribution

It proves Erdős–Szekeres type results for ordered matchings, determines bounds for twin structures, and generalizes findings to uniform and multiple twin matchings.

Findings

Every ordered matching contains a large basic sub-structure of lines, stacks, or waves.

Maximum size of twins in any ordered matching is between n^{3/5} and n^{2/3}.

Results extend to r-multiple twins and 3-uniform ordered matchings.

Abstract

An ordered matching of size $n$ is a graph on a linearly ordered vertex set $V$, $|V|=2n$, consisting of $n$ pairwise disjoint edges. There are three different ordered matchings of size two on $V=\{1,2,3,4\}$: an alignment $\{1,2\},\{3,4\}$, a nesting $\{1,4\},\{2,3\}$, and a crossing $\{1,3\},\{2,4\}$. Accordingly, there are three basic homogeneous types of ordered matchings (with all pairs of edges arranged in the same way) which we call, respectively, lines, stacks, and waves. We prove an Erd\H{o}s-Szekeres type result guaranteeing in every ordered matching of size $n$ the presence of one of the three basic sub-structures of a given size. In particular, one of them must be of size at least $n^{1/3}$. We also investigate the size of each of the three sub-structures in a random ordered matching. Additionally, the former result is generalized to $3$-uniform ordered matchings. Another type of unavoidable patterns we study are twins, that is, pairs of order-isomorphic, disjoint sub-matchings. By relating to a similar problem for permutations, we prove that the maximum size of twins that occur in every ordered matching of size $n$ is $O\left(n^{2/3}\right)$ and $\Omega\left(n^{3/5}\right)$. We conjecture that the upper bound is the correct order of magnitude and confirm it for almost all matchings. In fact, our results for twins are proved more generally for $r$-multiple twins, $r\ge2$.