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Furthermore, if each $H_i$ is what we call \\emph{homomorphism-critical} (in particular if each $H_i$ is a clique), then we determine $"},"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":"1705.01997","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-05-04T20:05:08Z","cross_cats_sorted":[],"title_canon_sha256":"ace9cd598d0f1c6b0757d235df80b3b4dc03e978c7a4fb0bb6120516f6ce2ab3","abstract_canon_sha256":"7b3645c6978353d521048a279456c55d94f509ac3786e21358e57cd7e61657dc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:11:11.742211Z","signature_b64":"P3W5c9m9oI1DnuNxgxvYIvIloeS4oyzQWe3MAmpOSmcuxqkZ6MTQ9NdeL2oh/eHQp5U7r9ZDuRXKreYdEtOsAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"748c0f0169877c655e1cb57900e85ed593f6d910df61f8d103f3b5a31ce1a1ad","last_reissued_at":"2026-05-18T00:11:11.741511Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:11:11.741511Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Edges not in any monochromatic copy of a fixed graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hong Liu, Maryam Sharifzadeh, Oleg Pikhurko","submitted_at":"2017-05-04T20:05:08Z","abstract_excerpt":"For a sequence $(H_i)_{i=1}^k$ of graphs, let $\\textrm{nim}(n;H_1,\\ldots, H_k)$ denote the maximum number of edges not contained in any monochromatic copy of $H_i$ in colour $i$, for any colour $i$, over all $k$-edge-colourings of~$K_n$.\n  When each $H_i$ is connected and non-bipartite, we introduce a variant of Ramsey number that determines the limit of $\\textrm{nim}(n;H_1,\\ldots, H_k)/{n\\choose 2}$ as $n\\to\\infty$ and prove the corresponding stability result. 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