A Lower Bound on the Size of Lipschitz Subsets in Dimension 3

A set S R d is C-Lipschitz in the x i-coordinate, where C > 0 is a real number, if for every two points a; b 2 S, we have ja i ? b i j C maxfja j ? b j j : j = 1; 2 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, Css ornyei, and Preiss. Here it is observed that a combinatorial result of Ruzsa and Szemer edi implies the existence of a 2-Lipschitz subset of size n 1=2 '(n) in every n-point set in R 3 , where '(n) ! 1 as n ! 1.