{"paper":{"title":"Extremal Betti Numbers and Persistence in Flag Complexes","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AT","math.OC"],"primary_cat":"math.CO","authors_text":"Lies Beers, Magnus Bakke Botnan","submitted_at":"2025-02-28T18:24:59Z","abstract_excerpt":"We investigate several problems concerning extremal Betti numbers and persistence in filtrations of flag complexes. For graphs on $n$ vertices, we show that $\\beta_k(X(G))$ is maximal when $G=\\mathcal{T}_{n,k+1}$, the Tur\\'an graph on $k+1$ partition classes, where $X(G)$ denotes the flag complex of $G$. Building on this, we construct an edgewise (one edge at a time) filtration $\\mathcal{G}=G_1\\subseteq \\cdots \\subseteq \\mathcal{T}_{n,k+1}$ for which $\\beta_k(X(G_i))$ is maximal for all graphs on $n$ vertices and $i$ edges. Moreover, the persistence barcode $\\mathcal{B}_k(X(G))$ achieves a max"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.21294","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/2502.21294/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"}