{"paper":{"title":"Greedy bases and relational complexity of diagonal type groups","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Primitive groups of diagonal type satisfy Cameron's conjecture on greedy base sizes and have relational complexity at least 4 that is unbounded.","cross_cats":[],"primary_cat":"math.GR","authors_text":"Colva M. Roney-Dougal, Hong Yi Huang","submitted_at":"2026-05-15T15:07:34Z","abstract_excerpt":"A base for a subgroup $G$ of $\\mathrm{Sym}(\\Omega)$ is a sequence of elements of $\\Omega$ with trivial pointwise stabiliser. The size of the smallest base for $G$ is denoted $b(G)$. There is a natural greedy algorithm to compute a base for $G$, and it was conjectured by Cameron in 1999 that there exists an absolute constant $c$ such that if $G$ is primitive then any base returned by this algorithm has size at most $cb(G)$. In this paper we determine the size of every base returned by the greedy algorithm when $G$ is a primitive group of diagonal type, and hence prove Cameron's conjecture for t"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We determine the size of every base returned by the greedy algorithm when G is a primitive group of diagonal type, and hence prove Cameron's conjecture for these groups. We prove that if G is primitive of diagonal type then RC(G) ≥ 4, that this lower bound is attained by infinitely many such G, and that the relational complexity of the groups of diagonal type is unbounded.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The analysis relies on the known structural description of primitive diagonal type groups (socle T^k acting on cosets of a diagonal subgroup) and on the fact that the greedy choice can be tracked via the orbits on the product space; if this structural description or the orbit calculations contain an undetected gap, the exact base sizes and the RC lower bound would not follow.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"For primitive diagonal type groups the greedy base sizes are computed exactly, proving Cameron's conjecture, while relational complexity is shown to be at least 4 with no upper bound.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Primitive groups of diagonal type satisfy Cameron's conjecture on greedy base sizes and have relational complexity at least 4 that is unbounded.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"826ac004653c026c1fcf90e4e6f423310a87dc2374a70ba2d0b772abe6927699"},"source":{"id":"2605.16032","kind":"arxiv","version":1},"verdict":{"id":"8776bfd6-2420-4be7-ba7f-637e2f934efa","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T18:58:50.233697Z","strongest_claim":"We determine the size of every base returned by the greedy algorithm when G is a primitive group of diagonal type, and hence prove Cameron's conjecture for these groups. We prove that if G is primitive of diagonal type then RC(G) ≥ 4, that this lower bound is attained by infinitely many such G, and that the relational complexity of the groups of diagonal type is unbounded.","one_line_summary":"For primitive diagonal type groups the greedy base sizes are computed exactly, proving Cameron's conjecture, while relational complexity is shown to be at least 4 with no upper bound.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The analysis relies on the known structural description of primitive diagonal type groups (socle T^k acting on cosets of a diagonal subgroup) and on the fact that the greedy choice can be tracked via the orbits on the product space; if this structural description or the orbit calculations contain an undetected gap, the exact base sizes and the RC lower bound would not follow.","pith_extraction_headline":"Primitive groups of diagonal type satisfy Cameron's conjecture on greedy base sizes and have relational complexity at least 4 that is unbounded."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.16032/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T19:31:19.011766Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T19:10:49.334956Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T17:33:41.565680Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T16:41:55.542451Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"1dc400e6dd78e1f96c89a0c1eb6efb3f90de8bd18d098fc9ac77f29e07e92576"},"references":{"count":31,"sample":[{"doi":"","year":2024,"title":"M. 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