Multiplicity covering number for the binary grid

Determine the correct lower bound for the minimum number of hyperplanes required to cover every point of the binary grid at least k times while missing one point.

Background

The paper reviews a multiplicity version of the Alon–Füredi hyperplane-covering theorem, in which every point of a finite grid must be covered at least k times while one point is omitted. A known lower bound is tight when one side of the grid is much larger than the others, but the authors note that its sharpness is unresolved for combinatorially important special cases, including the binary grid.

References

Although this bound is tight when |S1| is much larger than |Si| for all i # 2 (see the remark after the proof of Theorem 1.1 in [6]), in some combinatorially interesting special cases like the binary grid {0,1}? it is an open problem to determine the correct bound [11, 16].

— Covering half-grids with lines and planes  (2501.11156 - Bishnoi et al., 19 Jan 2025) in Section 1, paragraph discussing the multiplicity generalization of the Alon–Füredi theorem