Extremal, enumerative and probabilistic results on ordered hypergraph matchings
Michael Anastos, Zhihan Jin, Matthew Kwan, Benny Sudakov
TL;DR
This work develops a comprehensive framework for ordered $r$-uniform hypergraph matchings, introducing $r$-patterns and the notion of collectable patterns to extend the study of ordered matchings beyond the classical $r=2$ case. It establishes improved Ramsey-type bounds for the largest $P$-clique, resolves limiting behavior in random ordered matchings, and advances enumeration and extremal theory in this higher-dimensional setting, including exact extremal numbers for $r$-partite patterns. The authors combine poset methods, weak-pattern analysis, contraction/partitioning techniques, and probabilistic concentration to derive both upper and lower bounds, with several results sharpening known exponents (e.g., $L_r(n)$ bounds) and providing a rich set of open questions. The paper also outlines extensive directions for future work, including exact constants for small $r$, off-diagonal Ramsey variants, and deeper connections between extremal and enumerative aspects of ordered hypergraphs. Overall, it moves toward a fuller theory of ordered hypergraph matchings and highlights many intriguing avenues for further exploration in higher uniformities.
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 Ruciński 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.