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Let $\\pi^{(2)}(G)=\\lim_{n\\to\\infty}\\mathrm{ex}^{(2)}(n,G)/\\binom{n}{2}$ be the $2$-color Tur\\'{a}n density of $G$. What real numbers in the interval $(0,1)$ are realized as the $2$-color Tur\\'{a}n density of some graph? It is known that $\\pi^{(2)}(G)=1-(R_{\\chi}(G)-1)^{-1}$, where $R_{\\chi}(G)$ is the chromatic Ramsey number of $G$. Burr, Erd\\H{o}s, and Lov\\'{a}sz showed that $("},"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":"2409.07535","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-09-11T18:00:27Z","cross_cats_sorted":[],"title_canon_sha256":"ecfd8b527ba6a7b870a04e368ff6ea75dc7b6b19b6329d1f3c0b761f44bbb959","abstract_canon_sha256":"8b9ec4e246fdd8127b7a16cc9c67367f975eeefd5993f892007d23b5ca34a991"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-24T01:14:55.729796Z","signature_b64":"WicjaHxlgYznXV9eeI3YW+p06yZ7D/7XiqOxgYyksVk+IRZuglqPV1AQj8N8F5nJFeHuwfv6z1Px8LctLvJTCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d2a89b32503190d58e297c809d42bb01ca653683bd474f98bc476a93563cd1c3","last_reissued_at":"2026-06-24T01:14:55.729349Z","signature_status":"signed_v1","first_computed_at":"2026-06-24T01:14:55.729349Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Chromatic Ramsey numbers and two-color Tur\\'{a}n densities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dingyuan Liu, Maria Axenovich, Simon Gaa","submitted_at":"2024-09-11T18:00:27Z","abstract_excerpt":"Given a graph $G$, its $2$-color Tur\\'{a}n number $\\mathrm{ex}^{(2)}(n,G)$ is the maximum number of edges in an $n$-vertex graph, such that the edges can be colored with two colors avoiding a monochromatic copy of $G$. Let $\\pi^{(2)}(G)=\\lim_{n\\to\\infty}\\mathrm{ex}^{(2)}(n,G)/\\binom{n}{2}$ be the $2$-color Tur\\'{a}n density of $G$. What real numbers in the interval $(0,1)$ are realized as the $2$-color Tur\\'{a}n density of some graph? It is known that $\\pi^{(2)}(G)=1-(R_{\\chi}(G)-1)^{-1}$, where $R_{\\chi}(G)$ is the chromatic Ramsey number of $G$. 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