Calliope Ryan-Smith (Leeds)

Shattering Domination

The Erdős-Rado arrow relation is to the order property as the shattering domination relation is to the independence property. In attempting to create a faithful translation of the independence property in a first-order theory in the setting of abstract elementary classes, the problem of having no compactness becomes clear immediately. In a logical setting involving the compactness theorem, it is easy to find 'tree-indiscernible' sequences with the same EM-type as any arbitrary tree (as in the same way one can find order-indiscernible sequences with the same EM-type as any arbitrary sequence). However, without such tools, we are left with a much more blunt weapon: taking large, extant trees and finding within them structures that just so happen to be indiscernible. The shattering domination relation (and its numerous derivatives) is an attempt to measure how blunt that weapon is, that is to say how large a tree has to be before we can find an indiscernible sub-tree of a given size. In the setting of ZFC+GCH, this is solved, but it seems likely that in ZFC+¬GCH, it is independent.