Polynomial-order dichotomy in multidimensional saturation

Determine whether, for every multidimensional forbidden 0-1 pattern P, there exists an integer k such that sat(n,P)=Θ(n^k).

Background

For d-dimensional semisaturation, prior work shows that the growth rate is Θ(nr) for an integer exponent. For d-dimensional saturation, the paper notes only that the function is Ω(n) whenever it is not O(1), and the analogous polynomial-order classification is not known.

References

It is an open problem to determine if the same is true for the d-dimensional saturation function, though the same paper showed that the d-dimensional saturation function must be Ω(n) if it is not O(1).

— Saturation of 0-1 Matrices  (2503.03193 - Brahms et al., 5 Mar 2025) in Section 2.2 (p. 5)