In mathematics, the Herzog–Schönheim conjecture is a combinatorial problem in the area of group theory, posed by Marcel Herzog and Jochanan Schönheim in 1974.
be a group, and let be a finite system of left cosets of subgroups
Herzog and Schönheim conjectured that if
forms a partition of
In contrast, if repeated indices are allowed, then partitioning a group into cosets is easy: if
In 2004, Zhi-Wei Sun proved a special case of the Herzog–Schönheim conjecture in the case where
[2] A basic lemma in Sun's proof states that if
are subnormal and of finite index in
denotes the set of prime divisors of
of integers, the cosets of
In this case, the Herzog–Schönheim conjecture states that every covering system, a family of arithmetic progressions that together cover all the integers, must either cover some integers more than once or include at least one pair of progressions that have the same difference as each other.
This result was conjectured in 1950 by Paul Erdős and proved soon thereafter by Leon Mirsky and Donald J. Newman.
However, Mirsky and Newman never published their proof.
The same proof was also found independently by Harold Davenport and Richard Rado.
[3] In 1970, a geometric coloring problem equivalent to the Mirsky–Newman theorem was given in the Soviet mathematical olympiad: suppose that the vertices of a regular polygon are colored in such a way that every color class itself forms the vertices of a regular polygon.
Then, there exist two color classes that form congruent polygons.