Ordered asymptotic classes of finite structures
  In this paper we introduce the concept of O-asymptotic classes of finite structures, melding ideas coming from 1-dimensional asymptotic classes and o-minimality. The results we present here include a cell-decomposition result for O-asymptotic classes and a classification of their infinite ultraproducts: Every infinite ultraproduct of structures in an O-asymptotic class is superrosy of U-thorn-rank 1, and NTP2 (in fact, inp-minimal). 
Dar\'io Garc\'ia
math.LO