Title:  Linear order in NIP theories
Abstract: S. Shelah first defined the notion of NIP (No Independence Property) and proved that any unstable NIP theory has SOP (Strict Order Property) i.e. admits a pre-order with infinite chains. This is a fundamental result in the classification of first-order theories. 
This had raised the question of whether unstable NIP theories admit an infinite linear order. We will review Pierre Simon paper, where he proves a weaker version - in any unstable NIP theory one can find a V-definable equivalence relation, and a V-definable binary relation, such that a quotient by the equivalence relation makes the binary relation into a linear order with infinite chains. In the w-categorical case we even get a definable order. This result allows us to better understand such theories, as a linear order is a much more precise structure than partial order.