A lower bound on the size of Lipschitz subsets in dimension 3

Abstrakt

We prove: Any n-point set in R^3 has a subset of substantially more than n^{1/2} points 3-Lipschitz in some coordinate. A set S is C-Lipschitz in the z-coordinate if |z(a)-z(b)| < C max(|x(a)-x(b)|,|y(a)-y(b)|) for every a,b in S.