Minimally nonlinear sequences and 0-1 matrices

The length of every minimally nonlinear sequence with k distinct letters is at most (1+epsilon)4 k alpha(k) for all epsilon > 0.

The number of minimally nonlinear sequence with k distinct letters is at most sum_{i = 1}^{Ex(ababa,k)} (2k)^{i}.

The ratio between the number of rows and columns in a minimally nonlinear 0-1 matrix is between 0.25 and 4.

The number of minimally nonlinear 0-1 matrices with k rows is at most sum_{i = 1}^{4k-2} (2^{k}-1)^{i}.

Open questions:

What is the maximum number of ones in a minimally nonlinear 0-1 matrix with k rows?

What are the corresponding bounds for minimally nonlinear patterns in ordered graphs, bipartite ordered graphs, and separated bipartite ordered graphs?

Thread: http://www.artofproblemsolving.com/polymath/mitprimes2016/f/c195578h1320553_minimally_nonlinear_sequences_and_01_matrices