Bounds on Parameters of Minimally Nonlinear Patterns

Let ex(n, P ) be the maximum possible number of ones in any 0-1 matrix of dimensions n×n that avoids P . Matrix P is called minimally non-linear if ex(n, P ) 6= O(n) but ex(n, P ′) = O(n) for every proper subpattern P ′ of P . We prove that the ratio between the length and width of any minimally non-linear 0-1 matrix is at most 4, and that a minimally non-linear 0-1 matrix with k rows has at most 5k− 3 ones. We also obtain an upper bound on the number of minimally non-linear 0-1 matrices with k rows. In addition, we prove corresponding bounds for minimally non-linear ordered graphs. The minimal non-linearity that we investigate for ordered graphs is for the extremal function ex<(n,G), which is the maximum possible number of edges in any ordered graph on n vertices with no ordered subgraph isomorphic to G.