{"paper":{"title":"Fooling sets and rank","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC"],"primary_cat":"math.CO","authors_text":"Aya Hamed, Dirk Oliver Theis, Mirjam Friesen, Troy Lee","submitted_at":"2012-08-14T16:49:22Z","abstract_excerpt":"An $n\\times n$ matrix $M$ is called a \\textit{fooling-set matrix of size $n$} if its diagonal entries are nonzero and $M_{k,\\ell} M_{\\ell,k} = 0$ for every $k\\ne \\ell$. Dietzfelbinger, Hromkovi{\\v{c}}, and Schnitger (1996) showed that $n \\le (\\mbox{rk} M)^2$, regardless of over which field the rank is computed, and asked whether the exponent on $\\mbox{rk} M$ can be improved.\n  We settle this question. In characteristic zero, we construct an infinite family of rational fooling-set matrices with size $n = \\binom{\\mbox{rk} M+1}{2}$. In nonzero characteristic, we construct an infinite family of ma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.2920","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"}