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: Introduction to the KPT correspondence 3 (of 3).
Lecturer: Lionel Ngyuen Van Thé.
Date: November 18, 2016.
Definitions: Expansion property,
In the second lecture we saw that the Ramsey property of (a combinatorial property) ensures universality of a certain minimal flow (a dynamical property). Today we’ll look at going from a dynamical property (minimality) to a combinatorial property (the expansion property).
From last time
Recall that we proved the following in the second lecture:
Last time we saw that precompactness of the expansion allows us to topologically identify
We also saw that is a subset of a large compact product
Our main question today will be “What combinatorial properties guarantee that is a minimal flow?” More precisely, what condition must an expansion satisfy so that is minimal.
We say that has the expansion property (EP) (relative to ) when such that (expansions of respectively), we have embeds in .
When has the Joint Embedding Property, then (EP) is equivalent to such that (an expansion of ), we have embeds in .
The major theorem relating minimality and (EP)
Here is the major theorem we will prove.
- is minimal.
- has the (EP) relative to .
“You have to understand the purpose!” – Nešetřil.
“The difficulty is really translating into dynamical language what the combinatorics mean.” – Lionel
Before proving this theorem, we prove two propositions which will contain all the heavy lifting. For notational simplicity you may assume that is just a single relation .
- , for all .
- has the (EP) relative to .
“(2) is the correct finitization of (1).”
By (1), for all we have embeds into . So there is a finite such that
which is open in .
In this way forms an open cover of .
By compactness, there are finite such that .
Let be the finite substructure of supported by .
Claim: witnesses the (EP) for .
This is all that remains to finish the proof that .
This induces an embedding . By ultrahogeneity (for ) we can extend to an automorphism .
Then, for and we have
- , [Since preserves the structure, and extends .]
- , [by the definition of logic action].
So setting for all we get .
Now , so there is an such that .
Thus embeds into .
We now prove . Fix witnessing the (EP).
Take an . Then, by the (EP),
- is minimal,
- we have
- we have [Prop 1]
- has the (EP) relative to .
We can now combine this with the result from the second lecture (which tells us about universality) to get the following method for computing universal minimal flows.
This gives an explicit, combinatorial way to compute a universal minimal flow. You only need to find a precompact expansion of with (EP) and (RP). Often (RP) is used to prove (EP).
All of the universal minimal flows constructed in this way will be metrizable.
- For the following groups, the universal minimal flow is the logic action on the space : . [See lecture 2 for definitions of these spaces.]
- For the following groups, the universal minimal flow is the logic action on the space of “natural orders” (the ones that respect the linear orders “favoured” by the structure): (for a finite field).
- For , the universal minimal flow is the logic action on convex linear orders on .
“Converses” to the corollary
The following captures the uniqueness of a precompact expansion.
We saw in lecture 2 that the “smallness” of the universal minimal flow is dictated partly by the homogeneity and Ramsey properties of the group. The following theorem captures that notion.
- is metrizable with a orbit.
- admits a precompact expansion that is Fraïssé and has the (EP) and the (RP).
Why metrizability? It is a reasonable “smallness” condition.
This was expanded by Zucker, and he was able to drop the condition, while capturing the Ramsey degree.
- is metrizable.
- admits a precopact expansion that is Fraïssé and has the (EP) and the (RP).
- Every element in has a finite Ramsey degree.
That is, there is a such that colours appear on .
One way to interpret this result is that if you have a combinatorial property (3), then you get a precompact expansion with the (EP) and the (RP). This suggests (or at least seems to suggest) that precompact expansions are the relevant ones to consider.
Natural question (Tsankov 2009). Which satisfy these theorems? (Just knowing and not assuming (RP).)
Conjecture (Nguyen Van Thé 2012). When is precomapct.
This conjecture was shown to be false in 2015 by Evans using a Hrushovski construction. See his DocCourse lectures.
Conjecture (Bodirsky, Pinsker). This should be true for finite languages.
“What does the finite language mean topologically? Something about growth rate of number of structures of cardinality ? Related to amenability? Maybe the arity matters? This might require more examples of high arity.”
Openings from KPT
Research has gone in many directions from the original KPT paper.
- Continuous versions, metric Fraïssé. (See Dana Bartosova’s lectures later in the DocCourse.) This is useful to prove that some Polish groups are extremely amenable. e.g. the automorphism group of the Gurarij space, the homeomorphisms of the Poulsen space.
- KPT for other classes of things; i.e. universal minimal flow adjusted for distal or proximal flows. Melleray-Nguyen Van Thé-Tsankov 2016
- There are still some particular Ramsey problems for specific classes (still open after 10 years!) Euclidean metric spaces, and equidistributed Boolean algebras.
- Measures on . Which ones? How many? Angel-Kechris-Lyons 2014, Pawliuk-Sokic 2015.
- 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.
- Ngyuen Van Thé. “More on the Kechris–Pestov–Todorcevic correspondence: Precompact expansions”. 2013
- Ngyuen Van Thé. “A survey on structural Ramsey theory and topological dynamics with the Kechris-Pestov-Todorcevic correspondence in mind”. 2015
Other works cited (Mike: I have to fix some of these. This is obviously unfinished.)
- Dana – Guraji
- Dana – Poulsen
- Angel, Kechris, Lyons. “Random Orderings and Unique Ergodicity of Automorphism Groups.” 2014
- Pawliuk, Sokic. “Amenability and unique ergodicity of automorphism groups of countable homogeneous directed graphs.” [LINK] 2016
- Zucker. “Topological dynamics of automorphism groups, ultrafilter combinatorics, and the generic point problem.” 2016.
- Melleray, Nguyen Van Thé, Tsankov. “Polish groups with metrizable universal minimal flows.” 2016.
- Evans – Hrushovski