Bounds for subsets of 𝔽np×𝔽np without L-shaped configurations

I will discuss the difficult problem of proving reasonable bounds in the multidimensional generalization of Szemer\’edi’s theorem and describe a proof of such bounds for sets lacking nontrivial configurations of the form $(x,y), (x,y+z), (x,y+2z), (x+z,y)$ in the finite field model setting.