  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.
