{"paper":{"title":"Separation with restricted families of sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CO","authors_text":"G\\'abor Tardos, G\\'eza T\\'oth, J\\'anos Pach, M\\'arton Nasz\\'odi, Zsolt L\\'angi","submitted_at":"2015-08-22T12:51:22Z","abstract_excerpt":"Given a finite $n$-element set $X$, a family of subsets ${\\mathcal F}\\subset 2^X$ is said to separate $X$ if any two elements of $X$ are separated by at least one member of $\\mathcal F$. It is shown that if $|\\mathcal F|>2^{n-1}$, then one can select $\\lceil\\log n\\rceil+1$ members of $\\mathcal F$ that separate $X$. If $|\\mathcal F|\\ge \\alpha 2^n$ for some $0<\\alpha<1/2$, then $\\log n+O(\\log\\frac1{\\alpha}\\log\\log\\frac1{\\alpha})$ members of $\\mathcal F$ are always sufficient to separate all pairs of elements of $X$ that are separated by some member of $\\mathcal F$. This result is generalized to "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.05504","kind":"arxiv","version":1},"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"}