Adding linear orders

We address the following question: Can we expand an NIP theory by adding a linear order such that the expansion is still NIP? Easily, if acl(A)=A for all A, then this is true. Otherwise, we give counterexamples. More precisely, there is a totally categorical theory for which every expansion by a linear order has IP. There is also an ω-stable NDOP theory for which every expansion by a linear order interprets bounded arithmetic. A well known open question is whether every unstable NIP theory interprets an infinite linear order. We are concerned here with a question somewhat in the same spirit but going in a different direction: Can we expand an NIP theory by adding a linear order on the whole universe so that the resulting theory is still NIP? We give a negative answer in two strong forms: 1) There is an ω-stable NDOP theory of depth 2 for which every expansion by a linear order interprets bounded arithmetic (see section 2.2). 2) There is a totally categorical theory for which every expansion by a linear order has IP . In the first section, we mention a few positive statements that are true (and easy): if M is NIP and acl(A) = A for all A ⊂ M , then M can be linearly ordered so as to stay NIP . The same holds replacing both occurrences of NIP by ω-categorical (a well known fact) so we cannot expect to get the strong conclusion of 1) with a totally categorical theory. ∗The author would like to thank the Israel Science Foundation for partial support of this research (Grant no. 710/07). Publication 979 on Shelah’s list.