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### Cardinal invariants and model-theoretic tree properties

## Abstract

In Classification Theory, Shelah defined several cardinal invariants of a complete theory which detect the presence of certain trees among the definable sets, which in turn quantify the complexity of forking. In later model-theoretic developments, local versions of these invariants were recognized as marking important dividing lines - e.g. simplicity and NTP2. Around these dividing lines, a dichotomy theorem of Shelah states that a theory has the tree property if and only if it is witnessed in one of two extremal forms--the tree property of the first or second kind--and it was asked if there is a 'quantitative' analogue of this dichotomy in the form of a certain equation among these invariants. We will describe these model-theoretic invariants and explain why the quantitative version of the dichotomy fails, via a construction that relies upon some unexpected tools from combinatorial set theory.