High-rank subtensors of high-rank tensors

Let d ≥ 2 be a positive integer. We show that for a class of notions R of rank for order-d tensors, which includes in particular the tensor rank, the slice rank and the partition rank, there exist functions Fd,R and Gd,R such that if an order-d tensor has R-rank at least Gd,R(l) then we can restrict its entries to a product of sets X₁ × × Xd such that the restriction has R-rank at least l and the sets X₁, , Xd each have size at most Fd,R(l). Furthermore, our proof methods allow us to show that under a very natural condition we can require the sets X₁, , Xd to be pairwise disjoint.