Problem 274

If $G$ is a group then can there exist an exact covering of $G$ by more than one cosets of different sizes? (i.e. each element is contained in exactly one of the cosets)

Source: erdosproblems.com/274

Group theory
Covering systems

Wiki page

Status

Open

No claim settles this problem.

Er77c: Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72.
ErGr80: Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
MaSc19: Margolis, Leo and Schnabel, Ofir, The Herzog-Schönheim conjecture for small groups and harmonic subgroups. Beitr. Algebra Geom. (2019), 399-418.
Su04: Sun, Zhi-Wei, On the Herzog-Schönheim conjecture for uniform covers of groups. J. Algebra (2004), 153-175.

Problem 274

Status

Proof claims

Discussion