Lines in Euclidean Ramsey theory - Conlon, D; Fox, J (Springer US, 2018)
Let ℓm 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 En containing no red copy of ℓ2 and no blue copy of ℓm for any m≥2cn . 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⊂E1 and a red/blue-coloring of En containing no red copy of ℓ2 and no blue copy of K.