Determine the optimal multiplicity covering bound for the binary grid

Determine the correct minimum number of hyperplanes required to cover every point of the binary grid at least k times while missing one point, thereby resolving the multiplicity generalization of the Alon–Füredi theorem in this special case.

Background

The paper discusses the multiplicity version of the Alon–Füredi hyperplane-covering theorem, in which every grid point must be covered at least k times while one point is omitted. Ball and Serra established a general lower bound that is sharp when one side of the grid is much larger than the others. However, the authors note that the optimal value remains undetermined 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, Introduction