Borsuk and Ramsey Type Questions in Euclidean Space

We give a short survey of problems and results on (1) diameter graphs and hypergraphs, and (2) geometric Ramsey theory. We also make some modest contributions to both areas. Extending a well-known theorem of Kahn and Kalai that disproved Borsuk’s conjecture, we show that for any integer r ≥ 2, there exist ε = ε(r) > 0 and d ₀ = d ₀(r) with the following property. For every d ≥ d ₀, there is a finite point set of diameter 1 such that no matter how we color the elements of P with fewer than colors, we can always find r points of the same color, any two of which are at distance 1. Erdos, Graham, Montgomery, Rothschild, Spencer, and Strauss called a finite point set P Rd Ramsey if for every r ≥ 2, there exists a set RD for some D ≥ d such that no matter how we color all of its points with r colors, we can always find a monochromatic congruent copy of P. If such a set R exists with the additional property that its diameter is the same as the diameter of P, then we call P diameter-Ramsey. We prove that, in contrast to the original Ramsey property, (1) the condition that P is diameter- Ramsey is not hereditary, and (2) not all triangles are diameter-Ramsey. We raise several open questions related to this new concept.