Ample sets in Cartesian products

2026-07-04

AI Summary

This work extends the classical theory of ample sets from the Boolean hypercube to general Cartesian product spaces. By introducing subproducts and defining notions of shattering, projection, and strong projection on them, the paper proposes a novel definition of ample sets in Cartesian products and establishes several equivalent characterizations. The study uncovers deep connections between ample sets and their complements, isometric embeddings, and commutativity properties, and proves a decomposition theorem alongside the collapsibility of prism complexes. Combining tools from combinatorics, graph theory, and geometry, the framework not only generalizes existing theory but also yields new examples arising from gain graphs and quasi-median graphs, highlighting its broad applicability and rich topological structure.

Abstract

Ample sets of hypercubes, introduced by A. Dress in 1995, constitute a combinatorial structure with rich properties and important examples. Ample sets can be characterized in a multitude of combinatorial, graph-theoretical, recursive, and geometrical ways, and they are equivalent to lopsided sets introduced by J. Lawrence in 1983. In this paper, we define and investigate ample sets of Cartesian products $U=U_1\times\cdots\times U_m$. This is done using minor-subproducts of $U$, which correspond to products of partitions of factors: each minor-subproduct is obtained by partitioning each $U_i$ into blocks and contracting blocks into singletons. For a minor-subproduct $M$ and a set $S$, we define the notions of shattering of $M$ by $S$, of copy of $M$ in $S$, of projection $S_M$ of $S$ on $M$, and of strong-projection $S^M$ of $S$ on $M$. We call a set $S$ \emph{ample} if for any minor-subproduct $M$ that is shattered by $S$, there exists a copy of $M$ included in $S$. We prove that several characterizations of ample sets can be extended to ample sets of Cartesian products. In particular, we show that ampleness of $S$ is equivalent to the ampleness of the complement $S^*$, to superisometricity (isometricity of $S^M$ for any minor-subproduct $M$), and commutativity $(S^M)_{M'}=(S_{M'})^M$ for all minor-subproducts $M,M'$ with disjoint supports. We also provide more efficient characterizations of ampleness, in particular, by showing that $S$ is ample iff S is isometric and both $S_e$ and $S^e$ are ample for some elementary minor-subproduct, iff the intersection of S with any interval [u,v] with u,v in S is ample in the classical sense. We characterize ampleness by push downs and provide a decomposition theorem, allowing us to prove that their prism complexes are contractible. We provide new examples of ample sets arising from payoff games, prism-like polyhedra, and quasi-median graphs.

Problem

• ample sets
• Cartesian products
• minor-subproducts
• shattering
• isometricity

Innovation

• ample sets
• Cartesian products
• minor-subproducts
• superisometricity
• prism complexes