{"paper":{"title":"On small balanceable, strongly-balanceable and omnitonal graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Christina Zarb, Josef Lauri, Yair Caro","submitted_at":"2019-08-22T07:34:08Z","abstract_excerpt":"In Ramsey theory for graphs we are given a graph $G$ and we are required to find the least $n_0$ such that, for any $n\\geq n_0$, any red/blue colouring of the edges of $K_n$ gives a subgraph $G$ all of whose edges are blue or all are red. Here we shall be requiring that, for any red/blue colouring of the edges of $K_n$, there must be a copy of $G$ such that its edges are partitioned equally as red or blue (or the sizes of the colour classes differs by one in the case when $G$ has an odd number of edges). This introduces the notion of balanceable graphs and the balance number of $G$ which, if i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.08237","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.08237/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}