{"paper":{"title":"The Hajnal--Rothschild problem","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Andrey Kupavskii, Peter Frankl","submitted_at":"2025-02-10T17:24:56Z","abstract_excerpt":"For a family $\\mathcal F$ define $\\nu(\\mathcal F,t)$ as the largest $s$ for which there exist $A_1,\\ldots, A_{s}\\in \\mathcal F$ such that for $i\\ne j$ we have $|A_i\\cap A_j|< t$. What is the largest family $\\mathcal F\\subset{[n]\\choose k}$ with $\\nu(\\mathcal F,t)\\le s$? This question goes back to a paper Hajnal and Rothschild from 1973. We show that, for some absolute $C$ and $n>2k+Ct^{4/5}s^{1/5}(k-t)\\log_2^4n$, $n>2k+Cs(k-t)\\log_2^4 n$ the largest family with $\\nu(\\mathcal F,t)\\le s$ has the following structure: there are sets $X_1,\\ldots, X_s$ of sizes $t+2x_1,\\ldots, t+2x_s$, such that for"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.06699","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/2502.06699/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"}