{"paper":{"title":"Typical structure of hereditary graph families. I. Apex-free families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Sergey Norin, Yelena Yuditsky","submitted_at":"2020-07-01T18:10:11Z","abstract_excerpt":"A family of graphs $\\mathcal{F}$ is hereditary if $\\mathcal{F}$ is closed under isomorphism and taking induced subgraphs. The speed of $\\mathcal{F}$ is the sequence $\\{|\\mathcal{F}^n|\\}_{n \\in \\mathbb{N}}$, where $\\mathcal{F}^n$ denotes the set of graphs in $\\mathcal{F}$ with the vertex set $[n]$. Alon, Balogh, Bollob\\'{a}s and Morris [The structure of almost all graphs in a hereditary property, JCTB 2011] gave a rough description of typical graphs in a hereditary family and used it to show for every proper hereditary family $\\mathcal{F}$ there exist $\\varepsilon>0$ and an integer $l \\geq 1$ s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2007.00686","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/2007.00686/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"}