{"paper":{"title":"A disproof of the uniform witness conjecture","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Zixiang Xu","submitted_at":"2026-06-23T16:35:12Z","abstract_excerpt":"The study of $(d+1)$-uniform set systems with VC-dimension at most $d$ links the Erd\\H{o}s--Ko--Rado theorem with VC-dimension. But already in 1997, Ahlswede and Khachatrian showed that this is not the right extension of the Erd\\H{o}s--Ko--Rado theorem. In 2025, Chao, Xu, Yip and Zhang proposed the uniform witness conjecture as a possible right extension: for $0\\le s\\le d$, if every set of a $(d+1)$-uniform family has a missing trace of the same fixed size $s$, then the family should have size at most $\\binom{n-1}{d}$. They proved the conjecture when $s=d$, and when $s=1$ and $n$ is large. Ver"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.24776","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.24776/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"}