{"paper":{"title":"On clique-to-clique densities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jie Ma, Tianhen Wang, Tianming Zhu","submitted_at":"2026-06-30T17:10:55Z","abstract_excerpt":"Let $k_r(G)$ denote the number of $r$-cliques in a graph $G$ and let $F_r(\\cdot)$ be the Lov\\'asz--Simonovits $r$-clique density function. For any integers $2\\le s<t$, we determine the asymptotically sharp lower bound on $k_t(G)$ in an $n$-vertex graph $G$ with a prescribed number $k_s(G)$, by showing that \\[ \\frac{k_t(G)}{n^t}\\ge F_t\\!\\left(F_s^{-1}\\!\\left(\\frac{k_s(G)}{n^s}\\right)\\right), \\] where $F_s^{-1}$ denotes the generalized inverse. This strengthens Bollob\\'as's piecewise-linear interpolation bound and, in the case $s=2$, recovers Reiher's clique density theorem via a new inductive p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.31967","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/2606.31967/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"}