On subtensors of high partition rank

We prove that for every positive integer d ≥ 2 d ≥ 2 there exist polynomial functions F d , G d : N → N F_d, G_d: N → N such that for each positive integer r r , every order- d d tensor T T over an arbitrary field and with partition rank at least G d ( r ) G_d(r) contains a F d ( r ) × ⋯ × F d ( r ) F_d(r) × ⋯ × F_d(r) subtensor with partition rank at least r r . We then deduce analogous results on the Schmidt rank of polynomials in zero or high characteristic.