{"id":"palomar/PALOMAR-2026-09-25-000003","url":"https://github.com/Dishah3241/Erdos1007Dim5/tree/85032a43ca6db7d92b492efd05deeea0739c4570","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-09-25-000003-v1.json","repo":"https://github.com/Dishah3241/Erdos1007Dim5","tags":["math.CO","math.MG","msc:05C62","msc:52C10","graph-theory","euclidean-dimension","erdos-problems"],"score":0,"title":"Erdős 1007, dimension five: fifteen edges, attained by K6 and K1,3,3","trust":"high","domain":"math.CO","source":"palomar","authors":["Dishant Shah"],"reviews":0,"subject":"math.CO","summary":"The least number of edges of a graph of Euclidean dimension five (Erdős–Harary–Tutte) is 15, and both K6 and K1,3,3 attain it (Chaffee and Noble 2016). These are formal-conjectures' erdos_1007.variants.dimension_five and dimension_five_extremal. A graph has dimension n when n is the least dimension in which its vertices can be placed injectively with every edge a unit segment. The lower bound is Chaffee and Noble's Theorem 10: every graph with at most fourteen edges has such a placement in ℝ⁴. Its proof removes a vertex of degree at most three, joins that vertex's neighbours to one another,…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-09-25","reactions":{},"formalized":true,"formalizes":[{"title":"Dimension 4 and dimension 5 graphs with minimum edge set","identifier":"https://ajc.maths.uq.edu.au/pdf/64/ajc_v64_p327.pdf","relationship":"formalizes"}]}