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Erd\\H{o}s and S\\'os asked: is $f(n)$ unbounded?\n  N. Alon, D. J. Kleitman, C. Pomerance, M. Saks and P. Seymour proved upper and lower bounds in terms of the smallest non-divisor ($\\mbox{snd}$) of $n$. We refine the upper bound as follows: $$f (n) \\leq c \\log \\mbox{snd}\\ {n}.$$"},"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":"1706.05539","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-06-17T14:00:34Z","cross_cats_sorted":[],"title_canon_sha256":"9a543904f8b0bce94e2fbdb5de724d51ba7592e22f4d56fea27743c1b3f3240e","abstract_canon_sha256":"eea8d1155c9a9381cdce5c2d9257f862308cba4ca22f727537d7525fa4f80957"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:49:33.898389Z","signature_b64":"TO+z2pzCk1hCX5axGfelRTMGnSpuodsepS6keBV3yNIBRFEhd60T2wjev0XwjKLL0lAEz8Jvt/Ctq3AYknFsBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1d0a11822bb1dc61d21b93f8848449b86dff64a00868fa04537e6f8ffbb4cb8d","last_reissued_at":"2026-05-17T23:49:33.897907Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:49:33.897907Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On small $n$-uniform hypergraphs with positive discrepancy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Danila Cherkashin, Fedor Petrov","submitted_at":"2017-06-17T14:00:34Z","abstract_excerpt":"A two-coloring of the vertices $V$ of the hypergraph $H=(V, E)$ by red and blue has discrepancy $d$ if $d$ is the largest difference between the number of red and blue points in any edge. Let $f(n)$ be the fewest number of edges in an $n$-uniform hypergraph without a coloring with discrepancy $0$. Erd\\H{o}s and S\\'os asked: is $f(n)$ unbounded?\n  N. Alon, D. J. Kleitman, C. Pomerance, M. Saks and P. Seymour proved upper and lower bounds in terms of the smallest non-divisor ($\\mbox{snd}$) of $n$. We refine the upper bound as follows: $$f (n) \\leq c \\log \\mbox{snd}\\ {n}.$$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.05539","kind":"arxiv","version":2},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1706.05539","created_at":"2026-05-17T23:49:33.897971+00:00"},{"alias_kind":"arxiv_version","alias_value":"1706.05539v2","created_at":"2026-05-17T23:49:33.897971+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.05539","created_at":"2026-05-17T23:49:33.897971+00:00"},{"alias_kind":"pith_short_12","alias_value":"DUFBDARLWHOG","created_at":"2026-05-18T12:31:12.930513+00:00"},{"alias_kind":"pith_short_16","alias_value":"DUFBDARLWHOGDUQ3","created_at":"2026-05-18T12:31:12.930513+00:00"},{"alias_kind":"pith_short_8","alias_value":"DUFBDARL","created_at":"2026-05-18T12:31:12.930513+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/DUFBDARLWHOGDUQ3SP4IJBCJXB","json":"https://pith.science/pith/DUFBDARLWHOGDUQ3SP4IJBCJXB.json","graph_json":"https://pith.science/api/pith-number/DUFBDARLWHOGDUQ3SP4IJBCJXB/graph.json","events_json":"https://pith.science/api/pith-number/DUFBDARLWHOGDUQ3SP4IJBCJXB/events.json","paper":"https://pith.science/paper/DUFBDARL"},"agent_actions":{"view_html":"https://pith.science/pith/DUFBDARLWHOGDUQ3SP4IJBCJXB","download_json":"https://pith.science/pith/DUFBDARLWHOGDUQ3SP4IJBCJXB.json","view_paper":"https://pith.science/paper/DUFBDARL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1706.05539&json=true","fetch_graph":"https://pith.science/api/pith-number/DUFBDARLWHOGDUQ3SP4IJBCJXB/graph.json","fetch_events":"https://pith.science/api/pith-number/DUFBDARLWHOGDUQ3SP4IJBCJXB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DUFBDARLWHOGDUQ3SP4IJBCJXB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DUFBDARLWHOGDUQ3SP4IJBCJXB/action/storage_attestation","attest_author":"https://pith.science/pith/DUFBDARLWHOGDUQ3SP4IJBCJXB/action/author_attestation","sign_citation":"https://pith.science/pith/DUFBDARLWHOGDUQ3SP4IJBCJXB/action/citation_signature","submit_replication":"https://pith.science/pith/DUFBDARLWHOGDUQ3SP4IJBCJXB/action/replication_record"}},"created_at":"2026-05-17T23:49:33.897971+00:00","updated_at":"2026-05-17T23:49:33.897971+00:00"}