Homogeneous substructures in random ordered hyper-matchings

TL;DR

The paper analyzes the maximum size of sub-matchings (cliques) in random ordered $r$-uniform matchings where every pair of edges adheres to a prescribed set of patterns $\mathcal{P}$. It introduces and leverages the cube framework ($\mathcal{C}(P,\pi)$), shattered by mega-blocks and blow-ups, to derive sharp order-of-magnitude results for broad families, including all $r$-partite patterns and Dyck-pattern collections, as well as two-pattern and mismatched pairs. The authors blend probabilistic tools (Azuma–Hoeffding, Talagrand), combinatorial devices (posets, trace reconstruction), and explicit constructions (cubes, wave seeds) to obtain tight asymptotics: e.g., linear growth for $r$-partite patterns, $\Theta(n^{1/(r-t)})$ growth for cubes with $t$ refinements, and $\Theta(n^{1/r})$ for Dyck-pattern families. These results illuminate the rich spectrum of subpattern-cliques in random matchings, provide a unified toolkit for pattern-avoiding and reconstructible structures, and point to further open problems in higher-order pattern triplets and process-driven dynamics.

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 order-isomorphic to a member of $\mathcal{P}$. In this paper we are interested in the size of a largest $\mathcal{P}$-clique in a random ordered $r$-uniform matching selected uniformly from all such matchings on a fixed vertex set $[rn]$. We determine this size (up to multiplicative constants) for several sets $\mathcal{P}$, including all sets of size $|\mathcal{P}|\le2$, the set $\mathcal{R}^{(r)}$ of all $r$-partite patterns, as well as sets $\mathcal{P}$ enjoying a Boolean-like, symmetric structure.

Figures (8)

• Figure 1.1: Three distinct $2$-matchings of size two.
• Figure 1.2: Cliques of size $4$ corresponding to the three patterns in $\mathcal{P}^{(2)}$.
• Figure 3.1: The product $M_0\times M_1\times M_{2}$ from Example \ref{['product']}. Each of its eight $6$-vertex edges is depicted by a red-blue-red-blue-red path. There are four edges containing $\{1,2\}$ and four edges containing $\{3,4\}$.
• Figure 4.1: A 3-edge spanning three sets.
• Figure 6.1: A matching $H$ of size 3 and its copy in $\rm^{(3)}_{n}$.

Theorems & Definitions (45)

• Theorem 1.1: JCTB_paper
• Theorem 1.2: DGR-socks
• Theorem 2.1
• Corollary 2.2
• Theorem 2.3
• Definition
• Example 1
• Theorem 2.4
• Proposition 2.5
• Example 2