An extremal problem on crossing vectors

For positive integers w and k, two vectors A and B from Z w are called k-crossing if there are two coordinates i and j such that A [ i ] − B [ i ] ≥ k and B [ j ] − A [ j ] ≥ k . What is the maximum size of a family of pairwise 1-crossing and pairwise non-k-crossing vectors in Z w ? We state a conjecture that the answer is k w − 1 . We prove the conjecture for w ≤ 3 and provide weaker upper bounds for w ≥ 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.