{"paper":{"title":"An extremal problem on crossing vectors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Bartosz Walczak, Micha{\\l} Laso\\'n, Noah Streib, Piotr Micek, William T. Trotter","submitted_at":"2012-05-08T20:57:11Z","abstract_excerpt":"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 natur"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.1824","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}