{"paper":{"title":"Equivalences of promise compactness principles","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"When a pair of finite structures lacks an Olšák polymorphism, its promise compactness principle is equivalent to the ultrafilter principle over ZF.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bertalan Bodor","submitted_at":"2026-04-09T15:31:02Z","abstract_excerpt":"For a pair of finite relational structures $(\\mathfrak{A},\\mathfrak{B})$ such that $\\mathfrak{A}$ homomorphically maps to $\\mathfrak{B}$ we denote by $K_{(\\mathfrak{A},\\mathfrak{B})}$ the following statement: for all structures $\\mathfrak{I}$ with the same signature as $\\mathfrak{A}$ if all finite substructures of $\\mathfrak{I}$ homomorphically maps to $\\mathfrak{A}$ then $\\mathfrak{I}$ homomorphically maps to $\\mathfrak{B}$. In this article, we show that if $(\\mathfrak{A},\\mathfrak{B})$ has no Ol\\v{s}\\'{a}k polymorphism, then $K_{(\\mathfrak{A},\\mathfrak{B})}$ is equivalent to the ultrafilter "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"if (𝔄,𝔅) has no Olšák polymorphism, then K_(𝔄,𝔅) is equivalent to the ultrafilter principle over ZF. This includes the statements K_(K3,K5) and K_(H2,Hc) for all c≥2.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The pair (𝔄,𝔅) has no Olšák polymorphism (and 𝔄 homomorphically maps to 𝔅); the equivalence holds specifically under this algebraic condition on the finite structures.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"For structure pairs (A,B) without Olšák polymorphisms, the promise compactness K_(A,B) is equivalent to the ultrafilter principle over ZF, including K_(K3,K5) and K_(H2,Hc).","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"When a pair of finite structures lacks an Olšák polymorphism, its promise compactness principle is equivalent to the ultrafilter principle over ZF.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"f7ff22d5b82e5275f31ec67d3086afc01366d2c4da9902e514f46ae5f52baefb"},"source":{"id":"2604.08365","kind":"arxiv","version":2},"verdict":{"id":"38b14b5f-b706-4d57-9330-d630a1b11f9e","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T16:55:29.284482Z","strongest_claim":"if (𝔄,𝔅) has no Olšák polymorphism, then K_(𝔄,𝔅) is equivalent to the ultrafilter principle over ZF. This includes the statements K_(K3,K5) and K_(H2,Hc) for all c≥2.","one_line_summary":"For structure pairs (A,B) without Olšák polymorphisms, the promise compactness K_(A,B) is equivalent to the ultrafilter principle over ZF, including K_(K3,K5) and K_(H2,Hc).","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The pair (𝔄,𝔅) has no Olšák polymorphism (and 𝔄 homomorphically maps to 𝔅); the equivalence holds specifically under this algebraic condition on the finite structures.","pith_extraction_headline":"When a pair of finite structures lacks an Olšák polymorphism, its promise compactness principle is equivalent to the ultrafilter principle over ZF."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.08365/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"}