SCHWEITZER 2018 Problem 3

We call an $n\times n$ matrix well groomed if it only contains elements $0$ and $1$ , and it does not contain the submatrix $\begin{pmatrix} 1& 0 0 & 1 \end{pmatrix}.$ Show that there exists a constant $c>0$ such that every well groomed, $n\times n$ matrix contains a submatrix of size at least $cn\times cn$ such that all of the elements of the submatrix are equal. (A well groomed matrix may contain the submatrix $\begin{pmatrix} 0& 1 1 & 0 \end{pmatrix}.$ )