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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2502.06699","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-02-10T17:24:56Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"53a2c4991b406e34c2b10dda1689221979c105b461fa48760546b54ac55604f0","abstract_canon_sha256":"33af4a16701d8b9b850e20cc942b94afea123c041dbc2c605c55a9e918b3fc1e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:12:10.936290Z","signature_b64":"igEIm9rvbCOOmHuP6CtwD2pIgJCEWmh1vOz500WVYWc+dHVt1yRNf+G4D+HNP3GgyyGNagmUjWBFtV/qX5xDCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bdadc6f602a1d88e094212dd7b96ab902dece55829c716f57e4e7b6130f68edc","last_reissued_at":"2026-07-05T10:12:10.935841Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:12:10.935841Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2502.06699","created_at":"2026-07-05T10:12:10.935891+00:00"},{"alias_kind":"arxiv_version","alias_value":"2502.06699v1","created_at":"2026-07-05T10:12:10.935891+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2502.06699","created_at":"2026-07-05T10:12:10.935891+00:00"},{"alias_kind":"pith_short_12","alias_value":"XWW4N5QCUHMI","created_at":"2026-07-05T10:12:10.935891+00:00"},{"alias_kind":"pith_short_16","alias_value":"XWW4N5QCUHMI4CKC","created_at":"2026-07-05T10:12:10.935891+00:00"},{"alias_kind":"pith_short_8","alias_value":"XWW4N5QC","created_at":"2026-07-05T10:12:10.935891+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":3,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2606.12380","citing_title":"Forbidden Intersection Theorems for Matrix Spaces","ref_index":3,"is_internal_anchor":false},{"citing_arxiv_id":"2607.00318","citing_title":"A Complete Intersection Theorem for Large Permutation Groups","ref_index":25,"is_internal_anchor":false},{"citing_arxiv_id":"2605.04987","citing_title":"Matchings in permutations","ref_index":14,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/XWW4N5QCUHMI4CKCCLOXXFVLSA","json":"https://pith.science/pith/XWW4N5QCUHMI4CKCCLOXXFVLSA.json","graph_json":"https://pith.science/api/pith-number/XWW4N5QCUHMI4CKCCLOXXFVLSA/graph.json","events_json":"https://pith.science/api/pith-number/XWW4N5QCUHMI4CKCCLOXXFVLSA/events.json","paper":"https://pith.science/paper/XWW4N5QC"},"agent_actions":{"view_html":"https://pith.science/pith/XWW4N5QCUHMI4CKCCLOXXFVLSA","download_json":"https://pith.science/pith/XWW4N5QCUHMI4CKCCLOXXFVLSA.json","view_paper":"https://pith.science/paper/XWW4N5QC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2502.06699&json=true","fetch_graph":"https://pith.science/api/pith-number/XWW4N5QCUHMI4CKCCLOXXFVLSA/graph.json","fetch_events":"https://pith.science/api/pith-number/XWW4N5QCUHMI4CKCCLOXXFVLSA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XWW4N5QCUHMI4CKCCLOXXFVLSA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XWW4N5QCUHMI4CKCCLOXXFVLSA/action/storage_attestation","attest_author":"https://pith.science/pith/XWW4N5QCUHMI4CKCCLOXXFVLSA/action/author_attestation","sign_citation":"https://pith.science/pith/XWW4N5QCUHMI4CKCCLOXXFVLSA/action/citation_signature","submit_replication":"https://pith.science/pith/XWW4N5QCUHMI4CKCCLOXXFVLSA/action/replication_record"}},"created_at":"2026-07-05T10:12:10.935891+00:00","updated_at":"2026-07-05T10:12:10.935891+00:00"}