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It has been conjectured that every Steiner triple system, STS$(v)$, on $v$ points $(v>7)$ admits a zero-sum $3$-flow. We show that for every pair $(v,\\lambda)$, for which a triple system, TS$(v,\\lambda)$ exists, there exists one which has a zero-sum $3$-flow, except when $(v,\\lambda)\\in\\{(3,1), (4,2), (6,2), (7,1)\\}$ and except possibl"},"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":"1502.04096","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-02-13T19:31:51Z","cross_cats_sorted":[],"title_canon_sha256":"14305faafe595af6a6ad0b8cefb95f12f22271280ebc6d36d836ac76f1705012","abstract_canon_sha256":"29872bc823e5b580732f0dd848bc8cb15f5a810e1b32de996e26bca1ab3676ec"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:14:16.911882Z","signature_b64":"mD6S2hU1BcQhkTSCuIiJXgMOwinMaqp9HnYiOV/GFJC0UdQ+wfqRLx6B/9cs1UXZWdJ/Fm5tYi8X2fw+fH00Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a4aba4a3669343b786267b214681ddc472b4508a1678d96257c582e01c9f4bad","last_reissued_at":"2026-05-18T02:14:16.911123Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:14:16.911123Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Zero-sum flows for Steiner triple systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"A.C. Burgess, E. Mendelsohn, P. Danziger, S. Akbari","submitted_at":"2015-02-13T19:31:51Z","abstract_excerpt":"Given a $2$-$(v,k,\\lambda)$ design, $\\cal{S}=(X,\\cal{B})$, a {\\it zero-sum $n$-flow} of $\\cal{S}$ is a map $f: \\cal{B} \\longrightarrow \\{\\pm 1, \\ldots ,\\pm (n-1)\\}$ such that for any point $x\\in X$, the sum of $f$ around all the blocks incident with $x$ is zero. It has been conjectured that every Steiner triple system, STS$(v)$, on $v$ points $(v>7)$ admits a zero-sum $3$-flow. We show that for every pair $(v,\\lambda)$, for which a triple system, TS$(v,\\lambda)$ exists, there exists one which has a zero-sum $3$-flow, except when $(v,\\lambda)\\in\\{(3,1), (4,2), (6,2), (7,1)\\}$ and except possibl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.04096","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":"1502.04096","created_at":"2026-05-18T02:14:16.911269+00:00"},{"alias_kind":"arxiv_version","alias_value":"1502.04096v2","created_at":"2026-05-18T02:14:16.911269+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.04096","created_at":"2026-05-18T02:14:16.911269+00:00"},{"alias_kind":"pith_short_12","alias_value":"USV2JI3GSNB3","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_16","alias_value":"USV2JI3GSNB3PBRG","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_8","alias_value":"USV2JI3G","created_at":"2026-05-18T12:29:44.643036+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/USV2JI3GSNB3PBRGPMQUNAO5YR","json":"https://pith.science/pith/USV2JI3GSNB3PBRGPMQUNAO5YR.json","graph_json":"https://pith.science/api/pith-number/USV2JI3GSNB3PBRGPMQUNAO5YR/graph.json","events_json":"https://pith.science/api/pith-number/USV2JI3GSNB3PBRGPMQUNAO5YR/events.json","paper":"https://pith.science/paper/USV2JI3G"},"agent_actions":{"view_html":"https://pith.science/pith/USV2JI3GSNB3PBRGPMQUNAO5YR","download_json":"https://pith.science/pith/USV2JI3GSNB3PBRGPMQUNAO5YR.json","view_paper":"https://pith.science/paper/USV2JI3G","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1502.04096&json=true","fetch_graph":"https://pith.science/api/pith-number/USV2JI3GSNB3PBRGPMQUNAO5YR/graph.json","fetch_events":"https://pith.science/api/pith-number/USV2JI3GSNB3PBRGPMQUNAO5YR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/USV2JI3GSNB3PBRGPMQUNAO5YR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/USV2JI3GSNB3PBRGPMQUNAO5YR/action/storage_attestation","attest_author":"https://pith.science/pith/USV2JI3GSNB3PBRGPMQUNAO5YR/action/author_attestation","sign_citation":"https://pith.science/pith/USV2JI3GSNB3PBRGPMQUNAO5YR/action/citation_signature","submit_replication":"https://pith.science/pith/USV2JI3GSNB3PBRGPMQUNAO5YR/action/replication_record"}},"created_at":"2026-05-18T02:14:16.911269+00:00","updated_at":"2026-05-18T02:14:16.911269+00:00"}