Homogeneous substructures in random ordered uniform matchings

TL;DR

The paper studies Erdős-Szekeres-type questions for random ordered $r$-uniform matchings by introducing and analyzing $\mathcal{P}$-cliques, where every pair of edges realizes a pattern from $\mathcal{P}$. It establishes sharp order-of-magnitude results for various $\mathcal{P}$, including the linear-size case for $\mathcal{R}^{(r)}$ and a detailed dichotomy for pairs of patterns (harmonious vs mismatch vs non-collectable), as well as a sublinear bound for triples under structured conditions. A core contribution is a robust probabilistic toolkit combining Azuma-type concentration for random permutations, a generic first-moment method, poset-Dilworth arguments, pattern-avoiding permutation bounds, and the trace method, enabling unified upper and lower bounds across r-values and pattern classes. The results connect to geometric and graph-theoretic interpretations, such as random interval and permutation graphs, and provide a framework for Erdős-Szekeres-type phenomena in random hypergraph settings. Overall, the work advances understanding of how large homogeneous substructures emerge in random ordered hypergraphs and offers multiple avenues for future extension, including sequential processes and “clean” substructures.

Abstract

An ordered $r$-uniform matching of size $n$ is a collection of $n$ pairwise disjoint $r$-subsets of a linearly ordered set of $rn$ vertices. For $n=2$, such a matching is called an $r$-pattern, as it represents one of $\tfrac12\binom{2r}r$ ways two disjoint edges may intertwine. Given a set $\mathcal{P}$ of $r$-patterns, a $\mathcal{P}$-clique is a matching with all pairs of edges belonging to $\mathcal{P}$. In this paper we determine the order of magnitude of the size of a largest $\mathcal{P}$-clique in a random ordered $r$-uniform matching for several sets $\mathcal{P}$, including all sets of size $|\mathcal{P}|\le2$ and the set $\mathcal{R}^{(r)}$ of all $2^{r-1}$ $r$-partite $r$-patterns.

Figures (6)

• Figure 1: Three distinct $2$-matchings of size two.
• Figure 2: Cliques of size $4$ corresponding to the three patterns in $\mathcal{P}^{(2)}$.
• Figure 3: Intersecting rectangles defined by $ABB$$\overline{B}$$A$$\overline{AAB}$ and $ABB$$\overline{B}$$A$$\overline{ABA}$.
• Figure 4: A 3-edge spanning three sets.
• Figure 5: A matching $H$ of size 3 and its copy in $\mathbb{RM}^{(3)}_{n}$.

Theorems & Definitions (25)

• theorem 1
• theorem 2: Dudek, Grytczuk and Ruciński JCTB_paper
• theorem 3
• theorem 4
• proposition 1
• lemma 1: Frieze and Pittel FP or McDiarmid McDiarmid98
• proof : Theorem \ref{['thm:random_r-partite']}
• lemma 2
• proof
• proof : Theorem \ref{['thm:r=2']}(c)