Lines in Euclidean Ramsey Theory

Abstract

Let ℓ_m be a sequence of m points on a line with consecutive points of distance one. For every natural number n, we prove the existence of a red/blue-coloring of E^n containing no red copy of ℓ_2 and no blue copy of ℓ_m for any m ≥ 2^(cn). This is best possible up to the constant c in the exponent. It also answers a question of Erdős et al. (J Comb Theory Ser A 14:341–363, 1973). They asked if, for every natural number n, there is a set K ⊂ E^1 and a red/blue-coloring of E^n containing no red copy of ℓ_2 and no blue copy of K.

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