Chang's conjecture
Introduction In model theory, a branch of mathematical logic, '''Chang's conjecture''', attributed to Chen Chung Chang by , states that every model of type (ω2,ω1) for a countable language has an elementary submodel of type (ω1, ω). A model is of type (α,β) if it is of cardinality α and a unary relation is represented by a subset of cardinality β. The usual notation is .The axiom of constructibility implies that Chang's conjecture fails. Silver proved the consistency of Chang's conjecture from the consistency of an ω1-Erdős cardinal. Hans-Dieter Donder showed a weak version of the reverse implication: if CC is not only consistent but actually holds, then ω2 is ω1-Erdős in K.More generally, Chang's conjecture for two pairs (α,β), (γ,δ) of cardinals is the claimthat every model of type (α,β) for a countable language has an elementary submodel of type (γ,δ). The consistency of  was shown by Laver from the consistency of a huge cardinal.
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