Partitioning multi-dimensional sets in a small number of “uniform” parts

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 almost regular. The latter means that the ratio between the maximal and minimal non-zero degree of the left nodes is bounded by a constant and the same condition holds for the right nodes. Stated differently, every finite 2-dimensional set S ⊂ N 2 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. ve 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 and their projections of multi-dimensional finite sets.