{"paper":{"title":"Cross-intersecting families of vectors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"G\\'abor Tardos, J\\'anos Pach","submitted_at":"2014-05-12T15:29:35Z","abstract_excerpt":"Given a sequence of positive integers $p = (p_1, . . ., p_n)$, let $S_p$ denote the family of all sequences of positive integers $x = (x_1,...,x_n)$ such that $x_i \\le p_i$ for all $i$. Two families of sequences (or vectors), $A,B \\subseteq S_p$, are said to be $r$-cross-intersecting if no matter how we select $x \\in A$ and $y \\in B$, there are at least $r$ distinct indices $i$ such that $x_i = y_i$. We determine the maximum value of $|A|\\cdot|B|$ over all pairs of $r$- cross-intersecting families and characterize the extremal pairs for $r \\ge 1$, provided that $\\min p_i >r+1$. The case $\\min "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.2805","kind":"arxiv","version":2},"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"}