Forbidden subposet problems in the grid

Abstract For posets P and Q, extremal and saturation problems about weak and strong P-free subposets of Q have been studied mostly in the case Q is the Boolean poset Q n , the poset of all subsets of an n-element set ordered by inclusion. In this paper, we study some instances of the problem with Q being the grid, and its connections to the Boolean case and to the forbidden submatrix problem.