  We study the existence of infinite cliques in omega-automatic (hyper-)graphs.
It turns out that the situation is much nicer than in general uncountable
graphs, but not as nice as for automatic graphs.
  More specifically, we show that every uncountable omega-automatic graph
contains an uncountable co-context-free clique or anticlique, but not
necessarily a context-free (let alone regular) clique or anticlique. We also
show that uncountable omega-automatic ternary hypergraphs need not have
uncountable cliques or anticliques at all.
