Extremal, enumerative and probabilistic results on ordered hypergraph matchings

TL;DR

This paper explores extremal, enumerative, and probabilistic properties of ordered hypergraph matchings, extending known results from 2-matchings to higher uniformities and addressing gaps in the current theory.

Contribution

The paper significantly advances the theory of ordered r-matchings for r ≥ 3 by improving existing results and proposing new research directions.

Findings

Enhanced bounds for ordered hypergraph matchings

New combinatorial techniques for r ≥ 3 cases

Open questions on extremal properties

Abstract

An ordered $r$-matching is an $r$-uniform hypergraph matching equipped with an ordering on its vertices. These objects can be viewed as natural generalisations of $r$-dimensional orders. The theory of ordered 2-matchings is well-developed and has connections and applications to extremal and enumerative combinatorics, probability, and geometry. On the other hand, in the case $r \ge 3$ much less is known, largely due to a lack of powerful bijective tools. Recently, Dudek, Grytczuk and Ruci\'nski made some first steps towards a general theory of ordered $r$-matchings, and in this paper we substantially improve several of their results and introduce some new directions of study. Many intriguing open questions remain.