Ample sets in Cartesian products

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.