We consider existentially closed fields with several orderings, valuations, and p-valuations. We show that these structures are NTP2 of finite burden, but usually have the independence property. Mo...

We show that if T is any geometric theory having the NTP2 then the corresponding theories of lovely pairs of models of T and of H‐structures associated to T also have the NTP2. We also prove that if T is strong then the same two expansions of T are also strong.

We introduce the notion of hereditary G-compactness. We provide a sufficient condition for a poset to not be hereditarily G-compact, which we use to show that any linear order is not hereditarily G-compact. Assuming that a long-standing conjecture about unstable NIP theories holds, this implies that an NIP theory is hereditarily G-compact if and only if it is stable -categorical theories). We show that if G is definable over A in a hereditarily G-compact theory, then \. We also include a (...) brief survey of sufficient conditions for G-compactness, with particular focus on those which can be used to prove or disprove hereditary G-compactness for some theories. (shrink)

We prove a couple of results on NTP2 theories. First, we prove an amalgamation statement and deduce from it that the Lascar distance over extension bases is bounded by 2. This improves previous work of Ben Yaacov and Chernikov. We propose a line of investigation of NTP2 theories based on S1 ideals with amalgamation and ask some questions. We then define and study a class of groups with generically simple generics, generalizing NIP groups with generically stable generics.