In mathematics, the Beck–Fiala theorem is a major theorem in
discrepancy theory
In mathematics, discrepancy theory describes the deviation of a situation from the state one would like it to be in. It is also called the theory of irregularities of distribution. This refers to the theme of ''classical'' discrepancy theory, name ...
due to
József Beck
József Beck (Budapest, Hungary, February 14, 1952) is a Harold H. Martin Professor of Mathematics at Rutgers University.
His contributions to combinatorics include the partial colouring lemma and the Beck–Fiala theorem in '' discrepancy th ...
and
Tibor Fiala. Discrepancy is concerned with coloring elements of a ground set such that each set in a certain set system is as balanced as possible, i.e., has approximately the same number of elements of each color. The Beck–Fiala theorem is concerned with the case where each element doesn't appear many times across all sets. The theorem guarantees that if each element appears at most times, then the elements can be colored so that the imbalance is at most .
Statement
Formally, given a universe
:
and a collection of subsets
: