{"paper":{"title":"Definable combinatorics with dense linear orders","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Amit Kuber, Arihant Jain, Himanshu Shukla","submitted_at":"2018-07-16T20:34:53Z","abstract_excerpt":"We compute the model-theoretic Grothendieck ring, $K_0(\\mathcal{Q})$, of a dense linear order (DLO) with or without end points, $\\mathcal{Q}=(Q,<)$, as a structure of the signature $\\{<\\}$, and show that it is a quotient of the polynomial ring over $\\mathbb{Z}$ generated by $\\mathbb N_+\\times(Q\\sqcup\\{-\\infty\\})$ by an ideal that encodes multiplicative relations of pairs of generators. As a corollary we obtain that a DLO satisfies the pigeon hole principle (PHP) for definable subsets and definable bijections between them--a property that is too strong for many structures."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.06097","kind":"arxiv","version":1},"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"}