Summary: Partitioning multi-dimensional sets in a small
number of "uniform" parts
Ilan Newman Alexander Shen
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 non-zero 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 2-dimensional 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 non-empty horizontal section is bounded
by a constant and the same condition holds for vertical sections.
We prove a similar statement for n-dimensional 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