{"paper":{"title":"Two problems of Burr, Erd\\H os, Graham, and S\\'os on maximal anti-Ramsey functions for $P_4$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bo Ning, Mingze Li, Tianying Xie","submitted_at":"2026-06-29T16:09:26Z","abstract_excerpt":"Burr, Erd\\H os, Graham, and S\\'os introduced the maximal anti-Ramsey function $\\chi_{\\mathrm{S}}(n,e,L)$, the minimum number of colors required over all $n$-vertex graphs with at least $e$ edges such that every copy of $L$ is rainbow. In \\cite{BEGS1989}, they posed the following two problems: (i) Is it true that there exists $C>0$, such that for all $u\\ge 1$, $\\chi_{\\mathrm{S}}\\left(n,\\lfloor un \\rfloor,P_4 \\right)<Cu$ holds for all sufficiently large $n$? (ii) Is it true that for all $\\epsilon >0$, there exists $c(\\epsilon)>0$ such that for all sufficiently large $n$, \\\\ $\\chi_{\\mathrm{S}}\\le"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.30505","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/2606.30505/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"}