{"paper":{"title":"Van der Waerden type theorem for amenable groups and FC-groups","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.GR","authors_text":"Emilio Parini","submitted_at":"2024-11-24T21:38:06Z","abstract_excerpt":"We prove that for a discrete, countable, and amenable group $G$, if the direct product $G^2=G \\times G$ is finitely colored then $\\{ g \\in G : \\text{exists } (x,y) \\in G^2 \\text{ such that } \\{ (x,y),(xg,y),(xg,yg)\\} \\text{ is monochromatic} \\}$, is left IP$^{\\ast}$. This partially solves a conjecture of V. Bergelson and R. McCutcheon. Moreover, we prove that the result holds for $G^m$ if $G$ is an FC-group, i.e., all conjugacy classes of $G$ are finite."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.15987","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/2411.15987/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"}