Limitations of the slice rank method in additive combinatorics

The slice rank method gives exponential bounds for sets with no three-term arithmetic progression in finite vector spaces of odd characteristic and for three-sunflower-free families of subsets of a fixed ground set. We show that for $k\ge 4$, every tensor that is nonzero exactly on the $k$-term arithmetic progression relation or the $k$-sunflower relation has maximal slice rank over every coefficient field. When the support is prescribed only on pairwise distinct inputs, we obtain comparable lower bounds, which likewise rule out exponential savings.