The following notes are from the Ramsey DocCourse in Prague 2016. The notes are taken by me and I have edited them. In the process I may have introduced some errors; email me or comment below and I will happily fix them.

**Title**: Dual Ramsey, the Gurarij space and the Poulsen simplex 1 (of 3).

**Lecturer**: Dana Bartošová.

**Date**: December 12, 2016.

**Main Topics**: Comparison of various Fraïssé settings, metric Fraïssé definitions and properties, KPT of metric structures, Thick sets

**Definitions:** continuous logic, metric Fraïssé properties, NAP (near amalgamation property), PP (Polish Property), ARP (Approximate Ramsey Property), Thick, Thick partition regular.

Lecture 1 – Lecture 2 – Lecture 3

Ramsey DocCourse Prague 2016 Index of lectures.

## Introduction

Throughout the DocCourse we have primarily focused on Fraïssé limits of finite structures. As we saw in Solecki’s first lecture (not posted yet), it makes sense, and is useful, to consider Fraïssé limits in a broader context. Today we will discuss those other contexts.

Solecki’s first lecture discussed how to take *projective* Fraïssé limits. Panagiotopolous’ lecture (not posted yet) looked at a specific application of these projective limits. We will see how to take *metric* (direct) Fraïssé limits.

## Overview

Discrete | Compact | Metric Structure | |
---|---|---|---|

Size | Countable | Separable | Separable, complete |

Limit | Fraïssé limit | Quotient of the projective limit | (direct or projective) Metric Fraïssé limit |

Homogeneity | (ultra)homogeneity | Projective approximate homogeneity | Approximate homogeneity (*) |

Automorphism group | non-archimedian groups (closed subgroups of $S^\infty$ | homeomorphism groups | Polish Groups |

KPT, extremely amenable iff | RP | Dual Ramsey | Approximate RP (**) |

Metrizability of UMF iff | finite Ramsey degree | (***) | (Open) Compact RP? |

Where we’ve seen these | Classical | Solecki’s lectures | These lectures |

(*) – Exact homogeneity is often not possible.

(**) – In the projective setting this is fairly unexplored. These proofs are usually via direct (discrete) Ramsey, or through concentration of measure.

(***) – You have KPT before you take the quotient, but lose it after taking the quotient. e.g. UMF(pre-pseudo arc) is not metrizable (through RP). A question of Uspenskij asks about the UMF(pseudo arc).

## Continuous Logic definitions

In the context of Banach spaces, it makes sense to use continuous logic. This is where we instead of the usual $\{0,1\}$-valued logic we allow sentences to take on values in the interval $[0,1]$. We also suitably adjust the logical constructives.

Classical logic | Continuous logic |
---|---|

True | 0 |

False | 1 |

$=$ | $d$ |

$x \vee y$ | $\min\{x,y\}$ |

$x \wedge y$ | $\max\{x,y\}$ |

$\neg x$ | $1-x$ |

$x \Rightarrow y$ | $(x-y) \vee 0$ |

$\forall$ | $\sup$ |

$\exists$ | $\inf$ |

Now we define functions and relations. Let $(A,d)$ be a complete metric space. So $(A^n, d)$ will be given the sup metric.

- $F: A^n \rightarrow A$ comes with a Lipschitz constant.
- $R: A^n \rightarrow [0,1]$ comes with a Lipschitz constant.

Then functions and relations must satisfy the usual things that functions and relations satisfy in classical logic.

### Examples

Finitely generated substructures | Limit | maps | Language | |
---|---|---|---|---|

Separable metric spaces | finite metric spaces | Separable Urysohn space | isometric embedding | just the distance |

Separable Banach spaces | finite dimensional Banach spaces (**) | Gurarij space | isometric linear embedding | $\{|| \cdot ||, +, (\cdot \lambda)_{\lambda \in \mathbb{Q}}\}$ |

Separable Choquet spaces | finite dimensional simplices | Poulsen simplex | affine homeomorphisms (*) | Something that captures the convex structure |

(*) – An affine homeomorphism sends $S_0 \rightarrow S_1$ and sends extreme points to extreme points, then is extended affinely to the rest of the simplex. The metric here is not canonical.

(**) – Similar to the discrete case, to take a limit you only need a cofinal sequence. In this case we take $\ell^n_\infty$.

### Morphisms between models

In continuous logic the maps between models are isometric embeddings that preserves functions and relations.

## Properties of the finitely generated substructures

In the classical Fraïssé setting we looked at homogeneity, HP, JEP and AP. These notions have suitable generalizations in the metric Fraïssé setting.

**Definition**. Let $(A,d)$ be a metric structure. We describe finitely generated substructures in $(A,d)$ by $\langle \vec{a} \rangle$, where $\vec{a}$ is an $n$-tuple in $A$.

We say that $(A,d)$ is **approximately ultrahomogeneous (AUH)** if $\forall \vec{a} \in A^n, (\forall n)$ and for every $\phi: \langle \vec{a} \rangle \rightarrow A$ morphism, and for all $\epsilon >0$, there is a $\hat{\phi} \in \text{Aut}(A)$ such that $d(\phi(\vec{a}), \hat{\phi}(\vec{a}))<\epsilon$.

$\text{Age}(A)$ is the collection of finitely generated substructures of $A$.

**Lemma**. If $A$ is AUH and separable, then $\text{Age}(A)$ has

- HP,
- JEP,
- NAP (the Near Amalgamation Property),
- PP (the Polish Property, an analogue of countability).

We now explain NAP and PP. The NAP is a striaghtforward generalization of AP.

**Definition**. Let $\mathcal{K}$ be a collection of finitely generated metric structures. We say that $\mathcal{K}$ satisfies

**NAP**if when $f_i : A \rightarrow B_i$ are embeddings, then

$$\forall \epsilon > 0, \forall \vec{a} \in A^n, (\forall n), \exists C \in \mathcal{K}, \exists g_i : B_i \rightarrow C$$

such that

$$d_C (g_1 f_1 (\vec{a}), g_2 f_2 (\vec{a}) < \epsilon.$$

The PP measures how closely you can embed two metric spaces.

**Definition**. Let $\mathcal{K}$ be a collection of finitely generated metric structures with JEP. Define $K_n$ to be all pairs $(\vec{a}, A)$ where $\vec{a} \in A^n$ and $\langle \vec{a}\rangle = A$. Define $d_n$, a pseudometric on $K_n$ by

$$d_n((\vec{a}, A), (\vec{b}, B)) := \inf\{d_c(f(\vec{a}, g(\vec{b}))\},$$

where this is taken over al $C \in \mathcal{K}$ such that $A,B$ embed in $C$, and all embeddings $f: A \rightarrow C$, $g: B \rightarrow C$.

We say $\mathcal{K}$ satisfies the **Polish Property (PP)** if $(K_n, d_n)$ is separable for all $n$.

This gives us the following Fraïssé theorem for metric structures.

**Theorem (K. Schonetsonitis, I. BenYaacov)**. Let $A$ be a Polish structure. TFAE

- $A$ is AUH.
- $\text{Age}(A)$ satisfies HP, JEP, NAP and PP.

## The Urysohn space

Recall that $(\mathbb{U}, d)$ is the separable Urysohn space. It is the (unique) complete, separable metric space, universal for separable metric spaces and (exactly) ultrahomogeneous with respect to finite metric spaces.

Its age is the collection of finite metric spaces. It is a metric Fraïssé class.

Its automorphism group has a similar universal property.

**Theorem (Uspenskij)**. $\text{Iso}(\mathbb{U})$ is universal with respect to second countable topological groups.

See these notes for more information.

## Automorphism groups

Recall the following fact about (classical) Fraïssé structures.

**Theorem**. Every closed subgroup of $S^\infty$ can be represented as the automorphism group of a classical Fraïssé structure.

The following observation of Melleray is the corresponding fact for metric structures. It has a similar proof to the classical fact.

**Theorem(Melleray)**. Every Polish group $G$ can be represented as an automorphism group of an AUH relational Polish structure.

**Proof**. By a theorem of Uspenskij, $G$ is a closed subgroup of $\text{Iso}(\mathbb{U}, d)$. $G$ acts on $\mathbb{U}$ by isometries, so for all $n$, $G$ acts on $\mathbb{U}^n$ by the diagonal action.

For every orbit closure in $G$ of a point $x \in \mathbb{U}^n$ add a relational symbol $C = \overline{G \cdot c}$ called $R_C$.

## Extreme amenability

The first relevant result is the following:

**Theorem (Pestov 2002)**. $\text{Iso}(\mathbb{U}, d)$ is extremely amenable.

This proof uses the finite Ramsey theorem and concentration of measure.

**Theorem (KPT 2005)**. $\text{Iso}(\mathbb{U}_\mathbb{Q}, d, \leq)$ is extremely amenable and $\text{Iso}(\mathbb{U}, d) = \overline{\text{Iso}(\mathbb{U}_\mathbb{Q}, d)}$.

The KPT theorem for metric structures is given by the following.

**Theorem (Melleray, Tsankov)**. Let $A$ be AUH. TFAE:

- $\text{Aut}(A)$ is extremely amenable.
- $\text{Age}(A)$ satisfies ARP.

We define the approximate Ramsey Property.

**Definition**. Let $\mathcal{K}$ be a collection of finitely generated metric structures. For $A,B \in \mathcal{K}$, $\text{Emb}(A,B)$ is the collection of all morphisms from $A$ to $B$. There is a suitable distance between embeddings which we will not define here (in the special case of Banach spaces it is the operator norm).

(**ARP**):

$$\forall A,B \in \mathcal{K}, \forall r \geq 2, \forall \epsilon >0, \forall F \in [\text{Emb}(A,B)]^{<\omega},$$

there is a $C \in \mathcal{K}$ such that

$$\forall c: \text{Emb}(A,C) \rightarrow [r], \exists \phi \in \text{Emb}(B,C), \exists i \in [r]$$

such that

$$\{f \circ \phi : f \in F\} \subseteq (c^{-1}(i))_\epsilon.$$

Here $(X)_\epsilon \subset \text{Emb}(A,C)$, and the $\epsilon$-fattening is using the embedding distance (which we haven't defined).

Recall that in the infinite case, rigidity was needed to have the embedding RP. That is why in finite metric spaces we added linear orders to get the RP. However, metric spaces do satisfy the ARP (by Pestov from extreme amenabilty of $\text{Iso}(\mathbb{U},d)$, without needing to add linear orders.

Also, by using the usual compactness arguments, we can assume that the witness $C$ to ARP is the full Fraïssé limit.

## The stabilizer equivalence

In the KPT correspondence, we saw a useful connection between the stabilizer of a set and collections of finite structures. See Lionel Ngyuen van The’s first DocCourse lecture.

Here we mention an analogous connection.

**Definition**. Let $G = \text{Iso}(\mathbb{U},d)$. The neighbourhood given by $X \in [\mathbb{U}]^{ 0$ is denoted by

$$V_X^\epsilon := \{\phi \in \text{Iso}(\mathbb{U},d) : d(\phi(x),x) < \epsilon, \forall x \in X\}.$$

The pointwise stabilizer is

$$\text{Stab}(X) := \{\phi \in \text{Iso}(\mathbb{U},d) : \phi(x) = x, \forall x \in X\}.$$

The embedding distance, for $f,g \in \text{Emb}(X, \mathbb{U})$ is

$$d(f,g) = \max_{x \in X} \{d(f(x), g(x)\}.$$

**Correspondence**. $\text{Emb}(X, \mathbb{U})$ can be identified with $G / \text{Stab}(X)$.

**Proof**. If $f \in G / \text{Stab}(X)$, then $f : \mathbb{U} \rightarrow \mathbb{U}$ and $f \upharpoonright X \in \text{Emb}(X, \mathbb{U})$.

So we can reword the ARP for finite metric spaces, by transfering the colouring $c: \text{Emb}(A,\mathbb{U}) \rightarrow [r]$ to a colouring $\hat{c} : G / \text{Stab}(A) \rightarrow [r]$.

## Thick sets

Thickness is a group property that captures some Ramsey properties. This is desirable because we would like to be able to detect Ramsey type phenomena from the group itself, without having to know the underlying Fraïssé limit.

**Definition**. ($G = \text{Iso}(\mathbb{U},d)$). $T \subseteq G$ is

**thick**iff $$\forall V_X^\epsilon, \{g V_X^\epsilon T : g \in G\} \text{ has the finite intersection property.}$$

$G$ is **thick partition regular** iff $\forall V_X^\epsilon, \forall G / \text{Stab}(x) = \bigcup_{i=1}^n = P_i$ there is a $P_{i_0}$ that is thick.

**Theorem**. $G$ is thick partition regular iff $\text{Age}(\mathbb{U}, d)$ satisfies ARP.

This is really just unwinding definitions. Then by general topological dynamics abstract nonsense we get:

**Theorem**. If $G$ is thick partition regular then $G$ is extremely amenable.

Note that this is a theorem just about groups. This doesn’t use much of the structure of $\mathbb{U}$. Our goal is to prove extreme amenability without having to first prove Ramsey theorems.

## Next lectures

In the next lectures we will examine the Gurarij space and prove the ARP for $\ell_\infty^n$ (i.e. Banach spaces).

## References

(This is incomplete – Mike)

- Kechris, Pestov, Todorcevic. “Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups”. 2005.
- Pestov. “Dynamics of infinite-dimensional groups. The Ramsey-Dvoretzky-Milman phenomenon.”. 2006.