Does there exist a ramsey graph of C4 such that an induced subgraph has a monochromatic C4, no matter how the edges are colored?

4$\begingroup$ You need to write your question more carefully, with all the conditions. $\endgroup$– Brendan McKayMar 12 '20 at 1:05
If $m\gt n$ and $p\gt n\binom m2$ then the complete bipartite graph $K_{m,p}$ has the property that, in any edge coloring with at most $n$ colors, it has a monochromatic induced subgraph $K_{2,2}=C_4$.
I.e., if $n$ is finite, we can take $m=n+1$ and $p=n\binom{n+1}2+1$, while if $n=\aleph_\alpha$ we can take $m=\aleph_{\alpha+1}$ and $p=\aleph_{\alpha+2}$.