{"paper":{"title":"Counting odd cycles in locally dense graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Christian Reiher","submitted_at":"2016-04-22T22:48:14Z","abstract_excerpt":"We prove that for any given $\\varepsilon>0$ and $d\\in [0,1]$, every sufficiently large $(\\varepsilon, d)$-dense graph $G$ contains for each odd integer $r$ at least $(d^r-\\varepsilon)|V(G)|^r$ cycles of length $r$. Here, $G$ being $(\\varepsilon, d)$-dense means that every set $X$ containing at least~$\\varepsilon\\,|V(G)|$ vertices spans at least $\\tfrac d2\\, |X|^2$ edges, and what we really count is the number of homomorphisms from an $r$-cycle into $G$.\n  The result adresses a question of Y. Kohayakawa, B. Nagle, V. R\\\"odl, and M. Schacht."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.06833","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"}