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A note on uniform definability of types over finite sets in partial orders of finite width

Timo Krisam, Ori Segel

Abstract

In "VC density in some theories without the independence property" the authors asked whether any partial order of finite width has the VC1 property (i.e. every formula in one variable has UDTFS in one parameter). We give a negative answer and some related remarks.

A note on uniform definability of types over finite sets in partial orders of finite width

Abstract

In "VC density in some theories without the independence property" the authors asked whether any partial order of finite width has the VC1 property (i.e. every formula in one variable has UDTFS in one parameter). We give a negative answer and some related remarks.

Paper Structure

This paper contains 3 sections, 3 theorems, 4 equations.

Table of Contents

  1. Introduction and Preliminaries
  2. A counterexample of width 3
  3. Finite structures

Key Result

Lemma \oldthetheorem

Let $M$ a structure in some language $\mathcal{L}$ and $B\subseteq M^k$ some set. Let also $x$ a single variable, $y$ a variable of size $k$, $\psi\left(x\right)$ an $\mathcal{L}$-formula with $m$ paramaters from $B$ such that $\left|\psi\left(M\right)\right|\leq2^{d+1}-1$, $\varphi\left(x,y\right)$

Theorems & Definitions (21)

  • Definition \oldthetheorem
  • Definition \oldthetheorem: UDTFS
  • Definition \oldthetheorem: VCd
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Example \oldthetheorem
  • Claim
  • proof
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • ...and 11 more