For cycles and balanced complete multipartite graphs, every forest of copies of F is forced in large Ramsey graphs, but infinitely many graphs F admit Ramsey graphs that avoid all non-trivial such forests.
Łuczak,On Ramsey minimal graphs, Electron
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Unavoidable subgraphs in Ramsey graphs
For cycles and balanced complete multipartite graphs, every forest of copies of F is forced in large Ramsey graphs, but infinitely many graphs F admit Ramsey graphs that avoid all non-trivial such forests.