An \(n\)-by-\(m\) binary matrix \(M\) contains an \(n'\)-by-\(m'\) matrix \(P\) if there are integers \(1 \leq i_1 \leq \ldots \leq i_{n'} \leq n\) and \(1 \leq j_1 \leq \ldots \leq j_{m'} \leq m\) such that for all \(i' \in [n'], j' \in [m']\), \(P(i',j') = 1\) implies \(M(i_{i'},j_{j'}) = 1\). If \(M\) does not contain \(P\), then \(M\) avoids \(P\).

Given two integers \(n,m\), we denote \(Ex(P,n,m)\) as the maximum number of 1’s in a matrix avoiding a matrix \(P\). We also define \(Ex(P,n) = Ex(P,n,n)\).