{"paper":{"title":"Proofs of Two Conjectures of Alon on Subgraph Counts","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jiasheng Zeng, Peiru Kuang, Shuang Sun, Yan Wang","submitted_at":"2026-06-16T14:10:12Z","abstract_excerpt":"All graphs considered are finite with no isolated vertices. Let $N(m,H)$ be the maximum number of subgraphs of a graph $G$ isomorphic to $H$, taken over all graphs $G$ with $m$ edges. Alon proved that $N(m,H)=\\Theta_H(m^{\\gamma(H)})$, where $\\gamma(H)=(|V(H)|+D(H))/2$ and $D(H)=\\max_{S\\subseteq V(H)}(|S|-|N_H(S)|)$, and conjectured [Conjecture 1, Isr. J. Math., 1986] that limit of $N(m,H)/m^{\\gamma(H)}$ exists as $m\\to\\infty$. We prove this conjecture and identify the limit as $\\lambda(H)=\\Lambda(H)/|\\operatorname{Aut}(H)|$, where $\\Lambda(H)$ is characterized by a variational problem over fin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.18321","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/2606.18321/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"}