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We investigate some natural combinatorial principles related to the notion of mild ineffability, and use them to obtain new characterizations of mild ineffable and weakly compact cardinals. We also show that one of these principles may be satisfied by a successor cardinal. Finally, we establish a version for of the canonical Ramsey theorem for pairs.
We force from large cardinals a model of ZFC in which and both have the tree property. We also prove that if we strengthen the large cardinal assumptions, then in the final model even satisfies the super tree property.
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