Abstract: 

We introduce a framework that capturesmany known dividing lines in model theory, including stability, simplicity, NIP, NTP1, NTP2, NATP, NCTP, NBTP, NWP, NGP, and NPM^(k). It suggests a uniform way of characterizing dividing lines using consistency-inconsistency patterns and generalized indiscernibles. We show that witnesses to the failure of these dividing lines canbe chosen so that their consistent part has a realization of full algebraic dimension. We call thisresult the n-variable theorem and will explain how it can be applied to preservation problems foroommany model-theoretic constructions. We will see that generic trivializations, lovely pair expansions, H-structure expansions, generic subspace

expansions of vector spaces, algebraically closed fields with distinguished subfields, and expansions by generic derivations preserve all thedividing lines listed above. We will also discusshow this framework can be extended to higher-arity dividing lines such as NOP_k, NIP_k, and NFOP_k. 

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