{"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. Furthermore, if each $H_i$ is what we call \\emph{homomorphism-critical} (in particular if each $H_i$ is a clique), then we determine $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.01997","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}