{"paper":{"title":"Ramsey properties for tilings in random graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Danni Peng, Lucas Arag\\~ao, Rafael Filipe, Rafael Miyazaki, Xinbu Cheng, Zhifei Yan","submitted_at":"2026-05-20T17:55:00Z","abstract_excerpt":"Let $mH$ be the graph formed by $m$ vertex-disjoint copies of a graph $H$. Let $G \\to (H)_r$ denote that, in any $r$-colouring of the edges of $G$, there exists a monochromatic copy of $H$. In 1975, Burr, Erd\\H{o}s, and Spencer showed that if $H$ is a graph on $k$ vertices whose independence number is $\\alpha$, then $K_n \\to (mH)_2$, where $m\\sim n/(2k-\\alpha)$, and that the $1/(2k-\\alpha)$ factor is best possible. In the 1990s, R\\\"{o}dl and Ruci\\'{n}ski proved that, for all but a few graphs~$H$, the threshold for the property $\\mathbb{G}(n,p) \\to (H)_r$ is $n^{-1/m_2(H)}$. In this paper, gene"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.21471","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/2605.21471/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"}