We prove that for every positive integer d ≥ 2 there exist polynomial functions Fd, Gd : N → N such that for each positive integer r, every order-d tensor T over an arbitrary field and with partition rank at least Gd(r) contains a Fd(r) × · · · × Fd(r) subtensor with partition rank at least r. We then deduce analogous results on the Schmidt rank of polynomials in zero or high characteristic.
Draisma et al. (Mon,) studied this question.