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New simple theories from hypergraph sequences

M. Malliaris, S. Shelah

Abstract

We develop a family of simple rank one theories built over quite arbitrary sequences of finite hypergraphs. (This extends an idea from the recent proof that Keisler's order has continuum many classes, however, the construction does not require familiarity with the earlier proof.) We prove a model-completion and quantifier-elimination result for theories in this family. We develop a combinatorial property which they share. We invoke regular ultrafilters to show the strength of this property, showing that any flexible ultrafilter which is good for the random graph is able to saturate such theories.

New simple theories from hypergraph sequences

Abstract

We develop a family of simple rank one theories built over quite arbitrary sequences of finite hypergraphs. (This extends an idea from the recent proof that Keisler's order has continuum many classes, however, the construction does not require familiarity with the earlier proof.) We prove a model-completion and quantifier-elimination result for theories in this family. We develop a combinatorial property which they share. We invoke regular ultrafilters to show the strength of this property, showing that any flexible ultrafilter which is good for the random graph is able to saturate such theories.

Paper Structure

This paper contains 4 sections, 6 theorems, 32 equations.

Table of Contents

  1. Templates and theories
  2. Model completion and quantifier elimination
  3. A combinatorial property
  4. A separation via flexibility

Key Result

Lemma 2.9

For every $m<\omega$, the following holds. Let If $M \models T^{m_*}$, $N \models T^{m_*+1}$ and $\varphi$ is a sentence of $\tau(T^{m})$ of length $\leq m$, then $M \models \varphi ~ \iff ~ N \models \varphi$.

Theorems & Definitions (38)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Remark 1.5
  • Definition 1.6
  • Definition 1.7
  • Definition 1.8
  • Definition 1.10
  • Example 1.12
  • ...and 28 more