Non-Homogenizable Classes of Finite Structures
  Homogenization is a powerful way of taming a class of finite structures with several interesting applications in different areas, from Ramsey theory in combinatorics to constraint satisfaction problems (CSPs) in computer science, through (finite) model theory. A few sufficient conditions for a class of finite structures to allow homogenization are known, and here we provide a necessary condition. This lets us show that certain natural classes are not homogenizable: 1) the class of locally consistent systems of linear equations over the two-element field or any finite Abelian group, and 2) the class of finite structures that forbid homomorphisms from a specific MSO-definable class of structures of treewidth two. In combination with known results, the first example shows that, up to pp-interpretability, the CSPs that are solvable by local consistency methods are distinguished from the rest by the fact that their classes of locally consistent instances are homogenizable. The second example shows that, for MSO-definable classes of forbidden patterns, treewidth one versus two is the dividing line to homogenizability. 
Albert Atserias and Szymon Toru\'nczyk
cs.LO math.CO math.LO