An extremal problem on crossing vectors

TL;DR

This paper investigates the maximum size of vector families with specific crossing properties in integer lattices, proving a conjecture for low dimensions and providing bounds and constructions for higher dimensions.

Contribution

It proves a conjecture on the maximum size of certain crossing vector families for dimensions up to three and offers bounds and constructions for higher dimensions.

Findings

Proved the conjecture for w ≤ 3.

Provided weaker upper bounds for w ≥ 4.

Constructed examples matching the conjectured maximum size.

Abstract

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 $w\leq 3$ and provide weaker upper bounds for $w\geq 4$. 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.