Black Boxes

by Shelah. [Sh:309]

We shall deal comprehensively with Black Boxes, the intention being that provably in ZFC we have a sequence of guesses of extra structure on small subsets, the guesses are pairwise with quite little interaction, are far but together are ``dense''. We first deal with the simplest case, were the existence comes from winning a game by just writing down the opponent's moves. We show how it help when instead orders we have trees with boundedly many levels, having freedom in the last. After this we quite systematically look at existence of black boxes, and make connection to non-saturation of natural ideals and diamonds on them.

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