Graham's boundedness conjecture for Euclidean Ramsey sets

A finite set XX in a Euclidean space is Ramsey if, for every kNk\in\mathbb{N}, some dimension contains a monochromatic congruent copy of XX under every kk-colouring. Graham's boundedness conjecture. For every k,nNk,n\in\mathbb{N}, there exists d=d(k,n)Nd=d(k,n)\in\mathbb{N} such that whenever XX is a Ramsey set of size at most nn in some Euclidean space, every kk-colouring of Rd\mathbb{R}^d contains a monochromatic congruent copy of XX. This is presented as an open boundedness problem for Ramsey sets.

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Primary source

Rahil Baber, Natalie Behague, Asier Calbet, David Ellis, Joshua Erde, Ron Gray, Maria-Romina Ivan, Barnabás Janzer, Robert Johnson, Luka Milićević, John Talbot, Ta Sheng Tan and Belinda Wickes, “A collection of open problems in celebration of Imre Leader's 60th birthday”, arXiv:2310.18163 (2023).

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