{"paper":{"title":"A limit law of almost $l$-partite graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Vera Koponen","submitted_at":"2012-04-11T14:05:04Z","abstract_excerpt":"For integers $l \\geq 2$, $d \\geq 1$ we study (undirected) graphs with vertices $1, ..., n$ such that the vertices can be partitioned into $l$ parts such that every vertex has at most $d$ neighbours in its own part. The set of all such graphs is denoted $\\mbP_n(l,d)$. We prove a labelled first-order limit law, i.e., for every first-order sentence $\\varphi$, the proportion of graphs in $\\mbP_n(l,d)$ that satisfy $\\varphi$ converges as $n \\to \\infty$. By combining this result with a result of Hundack, Pr\\\"omel and Steger \\cite{HPS} we also prove that if $1 \\leq s_1 \\leq ... \\leq s_l$ are integers"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.2454","kind":"arxiv","version":2},"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"}