{"paper":{"title":"The smallest one-realization of a given set","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Kaishun Wang, Kefeng Diao, Ping Zhao","submitted_at":"2011-06-30T01:44:37Z","abstract_excerpt":"For any set $S$ of positive integers, a mixed hypergraph ${\\cal H}$ is a realization of $S$ if its feasible set is $S$, furthermore, ${\\cal H}$ is a one-realization of $S$ if it is a realization of $S$ and each entry of its chromatic spectrum is either 0 or 1. Jiang et al. \\cite{Jiang} showed that the minimum number of vertices of realization of $\\{s,t\\}$ with $2\\leq s\\leq t-2$ is $2t-s$. Kr$\\acute{\\rm a}$l \\cite{Kral} proved that there exists a one-realization of $S$ with at most $|S|+2\\max{S}-\\min{S}$ vertices. In this paper, we improve Kr$\\acute{\\rm a}$l's result, and determine the size of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.6099","kind":"arxiv","version":1},"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"}