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 F₃, ₑ and G₃, ₑ such that if an order-d tensor has R-rank at least G₃, ₑ (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 F₃, ₑ (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.