{"paper":{"title":"Exact Homomorphism Thresholds Beyond Cliques","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Chong Shangguan, Mingyuan Rong, Xinqi Huang","submitted_at":"2026-07-30T14:06:22Z","abstract_excerpt":"The chromatic threshold, originating in a question of Erd\\H{o}s and Simonovits, asks when a linear minimum-degree condition forces bounded chromatic number in H-free graphs. Motivated by a question of Thomassen, the homomorphism threshold asks for the stronger conclusion that every such graph admits a homomorphism to an H-free graph of bounded order. Since the work of Goddard and Lyle determined the clique case, exact homomorphism thresholds for individual non-complete forbidden graphs have remained unknown. In this paper, we extend the clique case to a larger family of forbidden graphs, deter"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.28241","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/2607.28241/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"}