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Dec 9 at 9:11 comment added bof I may be wrong but I think I've proved that the Rado graph works, see my new answer.
Dec 3 at 8:41 vote accept Dominic van der Zypen
Dec 3 at 6:17 comment added Fedor Petrov Very nice example. Is it possible to construct a graph with countable many vertices so that works for $n=\omega$, i.e., every maximal clique is infinite and the hypergraph of maximal cliques is not properly colorable with finitely many colors?
Dec 3 at 3:59 history answered bof CC BY-SA 4.0