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A $1$-plex is also known as a transversal.\n  It is well known that if $n$ is even then $B_n$, the addition table for the integers modulo $n$, possesses no transversals. We show that there are a great many latin squares that are similar to $B_n$ and have no transversal. As a consequence, the number of species of transversal-free latin squares is shown to be at least $n^{n^{3/2}(1/2-o(1))}$ for even $n\\rightarrow\\infty$.\n  We also"},"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":"1609.03001","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-09-10T05:14:25Z","cross_cats_sorted":[],"title_canon_sha256":"02e7bc4f66ad44afa0a198eb0d27e70b36a5c76b6829addb8d137df37797077e","abstract_canon_sha256":"b6cb8d01a0d4309f59dda1062c06720555907ee35129fe0928250820480eaded"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:26:30.696993Z","signature_b64":"Pwz68w9SzQsLvLH9hIdw44dfYZIBtZ5F9GK+s1+Als/cmlUVTtCDo1St33RDKnnJRI7EkHxVOdcauAk20jegBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"80c3e73f833f2e0b810f0e30a055cd22e4465f15780abd0b243cd5f39f47e13e","last_reissued_at":"2026-05-18T00:26:30.696158Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:26:30.696158Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Latin squares with no transversals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ian M. Wanless, Nicholas J. Cavenagh","submitted_at":"2016-09-10T05:14:25Z","abstract_excerpt":"A $k$-plex in a latin square of order $n$ is a selection of $kn$ entries that includes $k$ representatives from each row and column and $k$ occurrences of each symbol. A $1$-plex is also known as a transversal.\n  It is well known that if $n$ is even then $B_n$, the addition table for the integers modulo $n$, possesses no transversals. We show that there are a great many latin squares that are similar to $B_n$ and have no transversal. As a consequence, the number of species of transversal-free latin squares is shown to be at least $n^{n^{3/2}(1/2-o(1))}$ for even $n\\rightarrow\\infty$.\n  We also"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.03001","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":"1609.03001","created_at":"2026-05-18T00:26:30.696271+00:00"},{"alias_kind":"arxiv_version","alias_value":"1609.03001v2","created_at":"2026-05-18T00:26:30.696271+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.03001","created_at":"2026-05-18T00:26:30.696271+00:00"},{"alias_kind":"pith_short_12","alias_value":"QDB6OP4DH4XA","created_at":"2026-05-18T12:30:39.010887+00:00"},{"alias_kind":"pith_short_16","alias_value":"QDB6OP4DH4XAXAIP","created_at":"2026-05-18T12:30:39.010887+00:00"},{"alias_kind":"pith_short_8","alias_value":"QDB6OP4D","created_at":"2026-05-18T12:30:39.010887+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/QDB6OP4DH4XAXAIPBYYKAVONEL","json":"https://pith.science/pith/QDB6OP4DH4XAXAIPBYYKAVONEL.json","graph_json":"https://pith.science/api/pith-number/QDB6OP4DH4XAXAIPBYYKAVONEL/graph.json","events_json":"https://pith.science/api/pith-number/QDB6OP4DH4XAXAIPBYYKAVONEL/events.json","paper":"https://pith.science/paper/QDB6OP4D"},"agent_actions":{"view_html":"https://pith.science/pith/QDB6OP4DH4XAXAIPBYYKAVONEL","download_json":"https://pith.science/pith/QDB6OP4DH4XAXAIPBYYKAVONEL.json","view_paper":"https://pith.science/paper/QDB6OP4D","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1609.03001&json=true","fetch_graph":"https://pith.science/api/pith-number/QDB6OP4DH4XAXAIPBYYKAVONEL/graph.json","fetch_events":"https://pith.science/api/pith-number/QDB6OP4DH4XAXAIPBYYKAVONEL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QDB6OP4DH4XAXAIPBYYKAVONEL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QDB6OP4DH4XAXAIPBYYKAVONEL/action/storage_attestation","attest_author":"https://pith.science/pith/QDB6OP4DH4XAXAIPBYYKAVONEL/action/author_attestation","sign_citation":"https://pith.science/pith/QDB6OP4DH4XAXAIPBYYKAVONEL/action/citation_signature","submit_replication":"https://pith.science/pith/QDB6OP4DH4XAXAIPBYYKAVONEL/action/replication_record"}},"created_at":"2026-05-18T00:26:30.696271+00:00","updated_at":"2026-05-18T00:26:30.696271+00:00"}