Erdős Problem #274
Herzog-Schönheim conjecture
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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An exact covering of a group `G` is a finite collection of subgroups `{H_1, ..., H_k}` and representative `{g_1, ..., g_k}` such that the cosets `g_iH_i` are pairwise disjoint and their union covers `G`. Note that this differs from `Partition (α := Subgroup G)` because the covering condition there invokes `Subgroup.sup` which is subgroup generation and thus stronger than union. This definition is easier to use in this context than the alternative `Partition (α := Set G)`, which lacks subgroup de
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Proved, with a uniform exact bound and standalone verification, that every finite direct power \(A_5^r\) satisfies the Herzog–Schönheim conjecture; the general problem remains open.
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