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

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    $\begingroup$ You need to write your question more carefully, with all the conditions. $\endgroup$ Mar 12 '20 at 1:05
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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 $p=\aleph_{\alpha+2}$.

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