Lines in Euclidean Ramsey Theory

摘要

Let 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 En containing no red copy of 2 and no blue copy of m for any m2cn. This is best possible up to the constant c in the exponent. It also answers a question of Erds et al. (J Comb Theory Ser A 14:341-363, 1973). They asked if, for every natural number n, there is a set K subset of E1 and a red/blue-coloring of En containing no red copy of 2 and no blue copy of K.