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For positive integers $w$ and $k$, two vectors $A$ and $B$ from $mathbb{Z}^w$ are called $k$-crossing if there are two coordinates $i$ and $j$ such that $A[i]-B[i]geq k$ and $B[j]-A[j]geq k$. What is the maximum size of a family of pairwise $1$-crossing and pairwise non-$k$-crossing vectors in $mathbb{Z}^w$? We state a conjecture that the answer is $k^{w-1}$. We prove the conjecture for $wleq 3$ and provide weaker upper bounds for $wgeq 4$. Also, for all $k$ and $w$, we construct several quite different examples of families of desired size $k^{w-1}$. This research is motivated by a natural question concerning the width of the lattice of maximum antichains of a partially ordered set.
A spanning tree of an edge-colored graph is rainbow provided that each of its edges receives a distinct color. In this paper we consider the natural extremal problem of maximizing and minimizing the number of rainbow spanning trees in a graph $G$. Su
An extension of the well-known Szeged index was introduced recently, named as weighted Szeged index ($textrm{sz}(G)$). This paper is devoted to characterizing the extremal trees and graphs of this new topological invariant. In particular, we proved t
The bloom of complex network study, in particular, with respect to scale-free ones, is considerably triggering the research of scale-free graph itself. Therefore, a great number of interesting results have been reported in the past, including bounds
Let $G$ be a 3-partite graph with $k$ vertices in each part and suppose that between any two parts, there is no cycle of length four. Fischer and Matouu{s}ek asked for the maximum number of triangles in such a graph. A simple construction involving a
Wilfs Sixth Unsolved Problem asks for any interesting properties of the set of partitions of integers for which the (nonzero) multiplicities of the parts are all different. We refer to these as emph{Wilf partitions}. Using $f(n)$ to denote the number