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 Fd, Gd: 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) Gd (r) contains a F d (r) × ⋯ × F d (r) Fd (r) Fd (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.