Extensions of the linear bound in the Füredi–Hajnal conjecture

We present two extensions of the linear bound, due to Marcus and Tardos, on the number of 1-entries in an n×n (0,1)-matrix avoiding a fixed permutation matrix. We first extend the linear bound to hypergraphs with ordered vertex sets and, using previous results of Klazar, we prove an exponential bound on the number of hypergraphs on n vertices which avoid a fixed permutation. This, in turn, solves various conjectures of Klazar as well as a conjecture of Brändén and Mansour. We then extend the original Füredi–Hajnal problem from ordinary matrices to d-dimensional matrices and show that the number of 1-entries in a d-dimensional (0,1)-matrix with side length n which avoids a d-dimensional permutation matrix is O(nd−1).