{"paper":{"title":"Supersaturation of induced even cycles in locally sparse graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Adam D\\v{z}avoronok, Alexander Mylet, Maria-Cristina Popa, Ole Gabsdil, Yinghan Andie Shao","submitted_at":"2026-08-05T15:57:19Z","abstract_excerpt":"A graph $\\Gamma$ is $(c,t)$-sparse for $c > 0$ and $t \\ge 1$ if for every pair of vertex subsets $A, B \\subseteq V(\\Gamma)$ with $|A|, |B| \\ge t$, the number of edges $e(A,B)$ between them satisfies $ e(A,B) \\le (1 - c)|A||B|$. In this paper, we prove that for every integer $\\ell\\ge2$, there are $\\varepsilon > 0, C, C' > 0$ such that if an $n$-vertex graph $\\Gamma$ is $(1-\\varepsilon,t)$-sparse for some $t$, and has at least $Ct^{1-1/\\ell}n^{1+1/\\ell}$ edges, then $\\Gamma$ contains at least $C'n^2t^{2\\ell-2}$ induced copies of $C_{2\\ell}$. This partially resolves a problem of Ding, Gao, Liu, L"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.04985","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2608.04985/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"}