{"paper":{"title":"The Hadwiger-Nelson problem with two forbidden distances","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dan Ismailescu, Geoffrey Exoo","submitted_at":"2018-05-15T22:32:23Z","abstract_excerpt":"In 1950 Edward Nelson asked the following simple-sounding question:\n  \\emph{How many colors are needed to color the Euclidean plane $\\mathbb{E}^2$ such that no two points distance $1$ apart are identically colored?}\n  We say that $1$ is a \\emph{forbidden} distance. For many years, we only knew that the answer was $4$, $5$, $6$, or $7$. In a recent breakthrough, de Grey \\cite{degrey} proved that at least five colors are necessary.\n  In this paper we consider a related problem in which we require \\emph{two} forbidden distances, $1$ and $d$. In other words, for a given positive number $d\\neq 1$, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.06055","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"}