{"paper":{"title":"Well quasi-order and atomicity for combinatorial structures under consecutive orders","license":"http://creativecommons.org/licenses/by/4.0/","headline":"A general framework decides well quasi-order and atomicity for avoidance sets of combinatorial structures under consecutive orders.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Nik Ru\\v{s}kuc, Victoria Ironmonger","submitted_at":"2025-10-02T09:51:54Z","abstract_excerpt":"We consider partially ordered sets of combinatorial structures under consecutive orders, meaning that two structures are related when one embeds in the other such that `consecutive' elements remain consecutive in the image. Given such a partially ordered set, we may ask decidability questions about its avoidance sets: subsets defined by a finite number of forbidden substructures. Two such questions ask, given a finite set of structures, whether its avoidance set is well quasi-ordered (i.e. contains no infinite antichains) or atomic (i.e. cannot be expressed as the union of two proper subsets)."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We will establish a general framework, which enables us to answer these problems for a wide class of combinatorial structures, including graphs, digraphs and collections of relations.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the consecutive embedding order on the structures permits the extension of recent approaches into a single general framework that covers the stated wide class without additional restrictions.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Establishes a general framework to answer decidability questions on well quasi-order and atomicity for avoidance sets in posets of combinatorial structures under consecutive orders, covering graphs, digraphs and relations.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A general framework decides well quasi-order and atomicity for avoidance sets of combinatorial structures under consecutive orders.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"1f34b7a27131989a17aa2e3c00a6a79ab6f23f0d2f15ec0a442de760090d5865"},"source":{"id":"2510.01852","kind":"arxiv","version":5},"verdict":{"id":"3aa790a1-e52a-418c-b93b-abf4af7f335f","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-18T10:48:28.366621Z","strongest_claim":"We will establish a general framework, which enables us to answer these problems for a wide class of combinatorial structures, including graphs, digraphs and collections of relations.","one_line_summary":"Establishes a general framework to answer decidability questions on well quasi-order and atomicity for avoidance sets in posets of combinatorial structures under consecutive orders, covering graphs, digraphs and relations.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the consecutive embedding order on the structures permits the extension of recent approaches into a single general framework that covers the stated wide class without additional restrictions.","pith_extraction_headline":"A general framework decides well quasi-order and atomicity for avoidance sets of combinatorial structures under consecutive orders."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2510.01852/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}