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Happy Families and Their Relatives

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Mathematical, Foundational and Computational Aspects of the Higher Infinite

We first recall the notion of $ happy$$ families$ as well as their combinatorial properties. Then we present some families which are related to $ happy$$ families$ and investigate $ Mathias$$ forcing$ restricted to these families. In the second part we show the relation between $it Mathias$$ forcing$ and the $ Ramsey$$ property$ and discuss the still open problem whether one can take $ Mathias’$$ inaccessible$ away. In the last part, we sketch Shelah’s construction of a model of ZFC in which there are exactly $ 70$$ happy$$ ultrafilters$.

This talk is part of the Isaac Newton Institute Seminar Series series.

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