March 6
Henry Towsner,
University of Pennsylvania
A Status Report on the Unknown Order Property
The 'unknown order property' is a combinatorial dividing line identified as part of the 'tame hypergraph regularity' program. Tame graph regularity tells us that we can understand some classic model theoretic dividing lines, like stability, NIP, and distality, as describing when binary relations can be roughly approximated by rectangles (with the different properties leading to different kinds of approximations). Tame hypergraph regularity looks at higher arity relations (ternary relations are usually sufficient) and tries to find combinatorial properties which tell us when a relation can be approximated by lower-arity information ('cylinder intersection sets', the higher-arity generalization of rectangles).
The unknown order property is one natural way to generalize stability to ternary relations, but has proven to be more complicated to understand - the name refers to the fact that it still has no known combinatorial characterization. We will describe the motivation for this notion and the current state of the art, including the original results by Terry and Wolf identifying key counterexamples to the unknown order property, and recent results by Chernikov and Towsner showing that there cannot be any combinatorial characterization analogous to the excluded half-graph characterization of stability, but also showing that a big family of natural examples must have this property