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arXiv · 1205.1824

An extremal problem on crossing vectors

Michał Lasoń·Piotr Micek·Noah Streib·William T. Trotter·Bartosz Walczak
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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.

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Michał Lasoń, Piotr Micek, Noah Streib, William T. Trotter, Bartosz Walczak. 2014-08-19. An extremal problem on crossing vectors. https://doi.org/10.1016/j.jcta.2014.07.009

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