{"paper":{"title":"Chromatic thresholds for linear equations and recurrence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hong Liu, Ningyuan Yang, Shengtong Zhang, Zhuo Wu","submitted_at":"2026-03-05T18:56:15Z","abstract_excerpt":"Motivated by classical problems in extremal graph theory, we study a chromatic analogue of Roth-type questions for linear equations over $\\mathbb F_p$. Given a homogeneous equation $\\mathcal L:\\sum_{i=1}^k c_i x_i=0$ with $k\\ge 3$, we study $\\mathcal L$-solution-free sets $A\\subseteq \\mathbb F_p$ through the chromatic number of the Cayley graph $\\mathsf{Cay}(\\mathbb F_p,A)$. We introduce the \\emph{chromatic threshold} $\\delta_\\chi(\\mathcal L)$, the minimum density that guarantees bounded chromatic number of $\\mathsf{Cay}(\\mathbb F_p,A)$ among all $\\mathcal L$-solution-free sets $A$, and determ"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2603.05490","kind":"arxiv","version":2},"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/2603.05490/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"}