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We conjecture that for every admissible $n$ there is an STS$(n)$ with a cross-free set of size $\\lfloor{n-3\\over 3}\\rfloor$ which if true, is best possible. We prove this conjecture for the case $n=18k+3$, constructing an STS$(18k+3)$ containing a cross-free set of size $6k$. We note that some of the $3$-bichromatic STSs, constructed by Colbourn, Dinitz and Rosa, have cross-free sets o"},"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":"1509.05527","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-18T07:42:32Z","cross_cats_sorted":[],"title_canon_sha256":"ad41681ae4efc31c093d486ff4660d98e319be5e9221f84061f329f87852df11","abstract_canon_sha256":"505ddc28ff9c5d87b6299e8623cdbe7aae029cad72b7d03c5ff5dcaca3f5b73b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:42.937024Z","signature_b64":"WcKq3dngagS6nbZwztaL5C2x+Rzi4flnBWLuictLLIYe6IuxYnO8jStQdhNDUGJJKxvnLwgDH5V7wzlwNBYpAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b689a9dd5325251b89538f6513df1ea89f0ac2a68327fc787f8e7afcefcd4323","last_reissued_at":"2026-05-18T01:32:42.936487Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:42.936487Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Large Cross-free sets in Steiner triple systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Andras Gyarfas","submitted_at":"2015-09-18T07:42:32Z","abstract_excerpt":"A {\\em cross-free} set of size $m$ in a Steiner triple system $(V,{\\cal{B}})$ is three pairwise disjoint $m$-element subsets $X_1,X_2,X_3\\subset V$ such that no $B\\in {\\cal{B}}$ intersects all the three $X_i$-s. We conjecture that for every admissible $n$ there is an STS$(n)$ with a cross-free set of size $\\lfloor{n-3\\over 3}\\rfloor$ which if true, is best possible. We prove this conjecture for the case $n=18k+3$, constructing an STS$(18k+3)$ containing a cross-free set of size $6k$. We note that some of the $3$-bichromatic STSs, constructed by Colbourn, Dinitz and Rosa, have cross-free sets o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.05527","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":""},"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":"1509.05527","created_at":"2026-05-18T01:32:42.936576+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.05527v1","created_at":"2026-05-18T01:32:42.936576+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.05527","created_at":"2026-05-18T01:32:42.936576+00:00"},{"alias_kind":"pith_short_12","alias_value":"W2E2TXKTEUSR","created_at":"2026-05-18T12:29:47.479230+00:00"},{"alias_kind":"pith_short_16","alias_value":"W2E2TXKTEUSRXCKT","created_at":"2026-05-18T12:29:47.479230+00:00"},{"alias_kind":"pith_short_8","alias_value":"W2E2TXKT","created_at":"2026-05-18T12:29:47.479230+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/W2E2TXKTEUSRXCKTR5SRHXY6VC","json":"https://pith.science/pith/W2E2TXKTEUSRXCKTR5SRHXY6VC.json","graph_json":"https://pith.science/api/pith-number/W2E2TXKTEUSRXCKTR5SRHXY6VC/graph.json","events_json":"https://pith.science/api/pith-number/W2E2TXKTEUSRXCKTR5SRHXY6VC/events.json","paper":"https://pith.science/paper/W2E2TXKT"},"agent_actions":{"view_html":"https://pith.science/pith/W2E2TXKTEUSRXCKTR5SRHXY6VC","download_json":"https://pith.science/pith/W2E2TXKTEUSRXCKTR5SRHXY6VC.json","view_paper":"https://pith.science/paper/W2E2TXKT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.05527&json=true","fetch_graph":"https://pith.science/api/pith-number/W2E2TXKTEUSRXCKTR5SRHXY6VC/graph.json","fetch_events":"https://pith.science/api/pith-number/W2E2TXKTEUSRXCKTR5SRHXY6VC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/W2E2TXKTEUSRXCKTR5SRHXY6VC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/W2E2TXKTEUSRXCKTR5SRHXY6VC/action/storage_attestation","attest_author":"https://pith.science/pith/W2E2TXKTEUSRXCKTR5SRHXY6VC/action/author_attestation","sign_citation":"https://pith.science/pith/W2E2TXKTEUSRXCKTR5SRHXY6VC/action/citation_signature","submit_replication":"https://pith.science/pith/W2E2TXKTEUSRXCKTR5SRHXY6VC/action/replication_record"}},"created_at":"2026-05-18T01:32:42.936576+00:00","updated_at":"2026-05-18T01:32:42.936576+00:00"}