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

A set S subset of R-d is C-Lipschitz in the x(i)-coordinate, where C > 0 is a real number, if, for every two points a,b is an element of S, we have \a(i) - b(i)\ less than or equal to C max{\a(j) - b(j)\ : j = 1,2,...,d, j not equal i}. Motivated by a problem of Laczkovich, the author asked whether every n-point set in R-d contains a subset of size at least cn(1-1/d) that is C-Lipschitz in one of the coordinates, for suitable constants C and c > 0 (depending on d). This was answered negatively by Alberti, Csornyei and Preiss. Here it is observed that a combinatorial result of Ruzsa and Szemeredi implies the existence of a 2-Lipschitz subset of size n(1/2) phi(n) in every n-point set in R-3, where phi(n) --> infinity as n --> infinity.