Question:
Let m be given. Show that if n is large enough, then every n-by-n 0, 1-matrix has a principal submatrix of size m in which all elements above the diagonal are the same, and all elements below the diagonal are the same.
(How can we apply Ramsey's theorem to solve this problem on matrices? - this is the same question, I have a bit of a different approach to solving this and I was wondering if this is correct as well, (or not?):
My approach: We define a "special" matrix like this A s.t. $ a_{ij} = \begin{cases} 0 \text { if $i\ne j \land (u_i,u_j$) is blue} \\ 1 \text { if $i\ne j \land (u_j, u_i$) is red} \\ 0 \text{ if $i=j$} \end{cases}$ This matrix will indeed translate any clique to a 0-1 matrix as desired (with all equal values above and below the diagonal). Now all we need to show is that for every graph, a clique of size m exists (=a sub matrix of this design). But from Ramsey's theorem for some n we have: $R(m,m)=n<\infty$ therefore any large enough matrix (=some translation of any large enough graph) will have such a sub matrix.