title: Structure of null sets in Euclidean space. 
Some results and open problems

lecture at the conference "Real Analysis, Geometric Measure Theory, 
PDE and Banach Spaces". University of Warwick, August 17-19, 2007.

abstract: I will describe a decomposition theorem for sets of 
measure zero in the plane which can be deduced from an elementary 
combinatorial result (Dillworth's lemma). 
I will then outline a few applications to some elementary questions 
in Geometric Measure Theory, and in particular to the construction 
of non-differentiable Lipschitz functions. 
The extension of these results to higher dimensions is mostly open. 
This is joint work with D. Preiss (Warwick) and M. Cs\"ornyei 
(University College London).