{"paper":{"title":"Definable Combinatorics of Some Borel Equivalence Relations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Connor Meehan, William Chan","submitted_at":"2017-09-13T23:48:10Z","abstract_excerpt":"If $X$ is a set, $E$ is an equivalence relation on $X$, and $n \\in \\omega$, then define $$[X]^n_E = \\{(x_0, ..., x_{n - 1}) \\in {}^nX : (\\forall i,j)(i \\neq j \\Rightarrow \\neg(x_i \\ E \\ x_j))\\}.$$\n  For $n \\in \\omega$, a set $X$ has the $n$-J\\'onsson property if and only if for every function $f : [X]^n_= \\rightarrow X$, there exists some $Y \\subseteq X$ with $X$ and $Y$ in bijection so that $f[[Y]^n_=] \\neq X$. A set $X$ has the J\\'onsson property if and only for every function $f : (\\bigcup_{n \\in \\omega}[X]^n_=) \\rightarrow X$, there exists some $Y \\subseteq X$ with $X$ and $Y$ in bijection"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.04567","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"}