 
Summary: Partitioning multidimensional sets in a small
number of "uniform" parts
Noga Alon
Ilan Newman Alexander Shen
G´abor Tardos
Nikolai Vereshchagin
Abstract
Our main result implies the following easily formulated statement.
The set of edges E of every finite bipartite graph can be split into
poly(log E) subsets so that all the resulting bipartite graphs are al
most regular. The latter means that the ratio between the maximal
and minimal nonzero degree of the left nodes is bounded by a con
stant and the same condition holds for the right nodes. Stated differ
ently, every finite 2dimensional set S N2 can be partitioned into
poly(log S) parts so that in every part the ratio between the maximal
size and the minimal size of nonempty horizontal section is bounded
by a constant and the same condition holds for vertical sections.
We prove a similar statement for ndimensional sets for any n and
show how it can be used to relate information inequalities for Shannon
entropy of random variables to inequalities between sizes of sections
