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.
N. Hindman and D. Strauss had shown that, for discrete semigroups, the cartesian product of two central sets are central. They also proved that the product of J- sets and C-sets are also J-set and C-set and characterized when the infnite product of these sets are preserved. To prove these results they used the algebraic structure of Stone-Cech compactification of discrete semigroups. In this work we will give a combinatorial proof of the preserveness of those large sets under fnite cartesian product.
The general position number ${\rm gp}(G)$ of a connected graph $G$ is the cardinality of a largest set $S$ of vertices such that no three distinct vertices from $S$ lie on a common geodesic; such sets are refereed to as gp-sets of $G$. The general position number of cylinders $P_r\,\square\, C_s$ is deduced. It is proved that ${\rm gp}(C_r\,\square\, C_s)\in \{6,7\}$ whenever $r\ge s \ge 3$, $s\ne 4$, and $r\ge 6$. A probabilistic lower bound on the general position number of Cartesian graph powers is achieved. Along the way a formula for the number of gp-sets in $P_r\,\square\, P_s$, where $r…
Given a finite set $V$, a convexity $\mathscr{C}$, is a collection of subsets of $V$ that contains both the empty set and the set $V$ and is closed under intersections. The elements of $\mathscr{C}$ are called convex sets. The digital convexity, originally proposed as a tool for processing digital images, is defined as follows: a subset $S\subseteq V(G)$ is digitally convex if, for every $v\in V(G)$, we have $N[v]\subseteq N[S]$ implies $v\in S$. The number of cyclic binary strings with blocks of length at least $k$ is expressed as a linear recurrence relation for $k\geq 2$. A bijection is est…
The satisfiability problem for multilevel syllogistic extended with the Cartesian product operator (MLSC) is a long-standing open problem in computable set theory. For long, it was not excluded that such a problem were undecidable, due to its remarkable resemblance with the well-celebrated Hilbert's tenth problem, as it was deemed reasonable that union of disjoint sets and Cartesian product might somehow play the roles of integer addition and multiplication. To dispense with nonessential technical difficulties, we report here about a positive solution to the satisfiability problem for a slight…
We study configurations of $n$ points and $n$ lines that form $Θ(n^{4/3})$ incidences, when the point set is a Cartesian product. We prove structural properties of such configurations, such that there exist many families of parallel lines or many families of concurrent lines. We show that the line slopes have multiplicative structure or that many sets of $y$-intercepts have additive structure. We introduce the first infinite family of configurations with $Θ(n^{4/3})$ incidences. We also derive a new variant of a different structural point-line result of Elekes. Our techniques are based on the …
Let $A,B$ be disjoint sets of sizes $n$ and $m$. Let ${\mathcal Q}$ be a family of quadruples, having $2$ elements from $A$ and $2$ from $B$, such that any subset $S \subseteq A \cup B$ with $|S|=7$, $|S \cap A| \geq 2$ and $|S \cap B| \geq 2$ contains one of the quadruples. We prove that the smallest size of ${\mathcal Q}$ is $(1/16 + O(1/n) + O(1/m)) n^2 m^2$ as $n,m\to\infty$. We also solve asymptotically a more general two-partite Turán problem for quadruples.
Given a graph $G=(V(G), E(G))$ and a set $P\subseteq V(G)$, the following concepts have been recently introduced: $(i)$ two elements of $P$ are \emph{mutually visible} if there is a shortest path between them without further elements of $P$; $(ii)$ $P$ is a \emph{mutual-visibility set} if its elements are pairwise mutually visible; $(iii)$ the \emph{mutual-visibility number} of $G$ is the size of any largest mutual-visibility set. % In this work we continue to investigate about these concepts. We first focus on mutual-visibility in Cartesian products. For this purpose, too, we introduce and in…
Let $D=(V,E)$ be a strongly connected digraph and let $u ,v\in V(D)$. The maximum distance $md (u,v)$ is defined as\\ $md(u,v)$=max\{$\overrightarrow{d}(u,v), \overrightarrow{d}(v,u)$\} where $\overrightarrow{d}(u,v)$ denote the length of a shortest directed $u-v$ path in $D$. This is a metric. The boundary, contour, eccentric and peripheral sets of a strong digraph $D$ with respect to this metric have been defined, and the above said metrically defined sets of a large strong digraph $D$ have been investigated in terms of the factors in its prime factor decomposition with respect to Cartesian …
A graph is said to be well-edge-dominated if all its minimal edge dominating sets are minimum. It is known that every well-edge-dominated graph $G$ is also equimatchable, meaning that every maximal matching in $G$ is maximum. In this paper, we show that if $G$ is a connected, triangle-free, nonbipartite, well-edge-dominated graph, then $G$ is one of three graphs. We also characterize the well-edge-dominated split graphs and Cartesian products. In particular, we show that a connected Cartesian product $G\Box H$ is well-edge-dominated, where $G$ and $H$ have order at least $2$, if and only if $G…
We prove upper and lower bounds for the number of lines in general position that are rich in a Cartesian product point set. This disproves a conjecture of Solymosi and improves work of Elekes, Borenstein and Croot, and Amirkhanyan, Bush, Croot, and Pryby. The upper bounds are based on a version of the asymmetric Balog-Szemeredi-Gowers theorem for group actions combined with product theorems for the affine group. The lower bounds are based on a connection between rich lines in Cartesian product sets and amenability (or expanding families of graphs in the finite field case). As an application of…
We study the list chromatic number of the Cartesian product of any graph $G$ and a complete bipartite graph with partite sets of size $a$ and $b$, denoted $χ_\ell(G \square K_{a,b})$. We have two motivations. A classic result on the gap between list chromatic number and the chromatic number tells us $χ_\ell(K_{a,b}) = 1 + a$ if and only if $b \geq a^a$. Since $χ_\ell(K_{a,b}) \leq 1 + a$ for any $b \in \mathbb{N}$, this result tells us the values of $b$ for which $χ_\ell(K_{a,b})$ is as large as possible and far from $χ(K_{a,b})=2$. In this paper we seek to understand when $χ_\ell(G \square K_…
