Graham's boundedness conjecture for Euclidean Ramsey sets

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A finite set XX in a Euclidean space is Ramsey if, for every k∈Nk\in\mathbb{N}, some dimension contains a monochromatic congruent copy of XX under every kk-colouring. Graham's boundedness conjecture. For every k,n∈Nk,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.

References

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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