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Conlon and Fox (2019) demonstrated a coloring of $n$-dimensional Euclidean space avoiding red congruent copies of $\\ell_2$ and blue congruent copies of $K$ for $|K| > 10000^n\\log R$. We show here a stronger bound, that in fact $|K| > (11 + o(1))^n\\ln R$ suffices for arbitrary $1$-separated $K$, while the improvement $|K| > (5 + o(1))^n\\ln R$ holds in many cases, including when $K = \\ell_m$, or more generally when $K$ is contained"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2606.17194","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-15T18:34:02Z","cross_cats_sorted":[],"title_canon_sha256":"244cc56b6cd9dfc73ed0e0ef43271090b675b48cacd293fa618b43910c16b15b","abstract_canon_sha256":"dd6beb46b415746cf89f223a55e7c9c78628d81195283c541fa7a650ffa4222a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-19T16:10:05.574934Z","signature_b64":"ivcxCB6kNTCKL0veL9KKR8mk2CpM2Bo+jtSrfMqiGMpg26CYNTDU1KOsl3EJ9aoLLavTeMkjexUynjww3zPMCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"71ba820cd9edb854f60b7893713c74113334cdfce98f8733cf04f92b45f67c02","last_reissued_at":"2026-06-19T16:10:05.574600Z","signature_status":"signed_v1","first_computed_at":"2026-06-19T16:10:05.574600Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Improved bounds for lines and $1$-separated sets in Euclidean Ramsey theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Gabriel Currier, Jiaming Zhang, Param Mody, Zehan Xie","submitted_at":"2026-06-15T18:34:02Z","abstract_excerpt":"Let $K$ be a $1$-separated set of diameter at most $R-1$, and let $\\ell_m$ denote a collection of $m$ points on a line, with consecutive points of distance $1$ apart. 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