{"paper":{"title":"On the maximum number of integer colourings with forbidden monochromatic sums","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hong Liu, Katherine Staden, Maryam Sharifzadeh","submitted_at":"2017-09-27T15:50:28Z","abstract_excerpt":"Let $f(n,r)$ denote the maximum number of colourings of $A \\subseteq \\lbrace 1,\\ldots,n\\rbrace$ with $r$ colours such that each colour class is sum-free. Here, a sum is a subset $\\lbrace x,y,z\\rbrace$ such that $x+y=z$. We show that $f(n,2) = 2^{\\lceil n/2\\rceil}$, and describe the extremal subsets. Further, using linear optimisation, we asymptotically determine the logarithm of $f(n,r)$ for $r \\leq 5$. Similar results were obtained by H\\`an and Jim\\'enez in the setting of finite abelian groups."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.09589","kind":"arxiv","version":2},"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"}