{"paper":{"title":"Exchange properties of finite set-systems","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"D\\\"om\\\"ot\\\"or P\\'alv\\\"olgyi, J\\'anos Pach, Peter Frankl","submitted_at":"2021-03-26T09:57:17Z","abstract_excerpt":"In a recent breakthrough, Adiprasito, Avvakumov, and Karasev constructed a triangulation of the $n$-dimensional real projective space with a subexponential number of vertices. They reduced the problem to finding a small downward closed set-system $\\cal F$ covering an $n$-element ground set which satisfies the following condition: For any two disjoint members $A, B\\in\\cal F$, there exist $a\\in A$ and $b\\in B$ such that either $B\\cup\\{a\\}\\in\\cal F$ and $A\\cup\\{b\\}\\setminus\\{a\\}\\in\\cal F$, or $A\\cup\\{b\\}\\in\\cal F$ and $B\\cup\\{a\\}\\setminus\\{b\\}\\in\\cal F$. Denoting by $f(n)$ the smallest cardinalit"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2103.14358","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2103.14358/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}