This is an open problem. Vera Sos has asked at numerous conferences if Ramsey graphs (these are graphs on n vertices with the smallest possible homogeneous set size) are quasirandom, in that each ordered graph on k vertices appears with asymptotically the same frequency. A positive answer to Sos' question would give a positive answer to your question. Unfortunately, while we know some results about the structure of Ramsey graphs, these problems are still open.
Jacob Fox
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