Hypergraph sequences as a tool for saturation of ultrapowers

Abstract

Let T₁, T₂ be countable first-order theories, M_i ⊨ T_i, and 𝒟 any regular ultrafilter on λ ≥ ℵ₀. A longstanding open problem of Keisler asks when T₂ is more complex than T₁, as measured by the fact that for any such λ, 𝒟, if the ultrapower (M₂)^λ/𝒟 realizes all types over sets of size ≤ λ, then so must the ultrapower (M₁)^λ/𝒟. In this paper, building on the author's prior work [12] [13] [14], we show that the relative complexity of first-order theories in Keisler's sense is reflected in the relative graph-theoretic complexity of sequences of hypergraphs associated to formulas of the theory. After reviewing prior work on Keisler's order, we present the new construction in the context of ultrapowers, give various applications to the open question of the unstable classification, and investigate the interaction between theories and regularizing sets. We show that there is a minimum unstable theory, a minimum TP₂ theory, and that maximality is implied by the density of certain graph edges (between components arising from Szemerédi-regular decompositions) remaining bounded away from 0,1. We also introduce and discuss flexible ultrafilters, a relevant class of regular ultrafilters which reflect the sensitivity of certain unstable (non low) theories to the sizes of regularizing sets, and prove that any ultrafilter which saturates the minimal TP₂ theory is flexible.