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abstract elementary classes amalgamation Andrés Villaveces Applications of Model Theory Artem Chernikov Ayhan Günaydın biembeddability classification theory conference connected component definability of types de Finetti dividing DOP Eric Jaligot finite covers Forking Grothendieck Henson graphs invariant types John Goodrick large cardinals Lisbon Luminy MALOA Model theory NIP NTP1 NTP2 open question positions postdoc random graph SchröderBernstein property Shelah simple theories SOPn stability tree property VC densityMeta
Author Archives: Artem Chernikov
Model theoretic stability and definability of types, after A. Grothendieck
It had happened more than once that combinatorial model theoretic dividing lines introduced by Shelah were invented independently in different fields of mathematics. This curious note by Itai Ben Yaacov gives another example of this phenomenon:
We point out how the “Fundamental Theorem of Stability Theory”, namely the equivalence between the “non order property” and deﬁnability of types, proved by Shelah in the 1970s, is in fact an immediate consequence of Grothendieck’s “Critères de compacité” from 1952. The familiar forms for the deﬁning formulae then follow using Mazur’s Lemma regarding weak convergence in Banach spaces.
…
In a meeting in Kolkata in January 2013, the author asked the audience who had ﬁrst deﬁned the notion
of a stable formula and when, and to the expected answer replied that, no, it had been Grothendieck, in
the ﬁfties.
Posted in Classification theory, Model Theory
Tagged definability of types, Grothendieck, Shelah, stability
MALOA “final” conference in Luminy
I had just attended the final MALOA conference “Logic and interactions“. MALOA was a European network, which is now essentially finished (though the “final” meeting was not the last one, there will be a workshop in Manchester soon). The meeting took place at CIRM in Luminy, a wonderful place not only for mathematical reasons:
(you can see some more photos here).
As MALOA was about logic, rather than model theory, the topics of the talks were quite diverse, ranging from purely algebraic model theory (definable valuations and height bound in arithmetic nullstellensatz) to proof theory and even “logical description for behaviour analysis in aerospace systems”. Not sure how productive this diversity is, but at least it is entertaining.
Some talks that I found of particular interest are:
 Talks by Deirdre Haskell and Chris Laskowski on NIP, VCdensity and connections to probability and combinatorics (in general one could safely add to this list some more subjects including set theory). These all are quite fascinating topics which deserve some postings in the future. There are already some examples of importing ideas from combinatorics (e.g. the beautiful (p,q)theorem of Alon, Kleitman and Matousek) to prove modeltheoretic results (e.g. UDTFS for NIP theories), but I believe that many more connections remain to be discovered.
 Todor Tsankov spoke about generalizations of de Finetti’s theorem. Classical de Finetti’s theorem from probability theory says that a sequence of random variables is exchangeable if and only if it is independent and identically distributed over its tail sigmaalgebra. Various multidimensional generalizations of this characterization form the socalled exchangeability theory. This theorem can viewed as providing a classification of all probability measures on invariant under the action of . Now, in a general situation, given a permutation group acting on a countable set , one can’t really hope to give any kind of “classification” of invariant measures on as we are in the context of the general ergodic theory. However, it appears that if the group is sufficiently large compared to the index set, one can arrive at stronger results. Todor’s approach is to consider oligomorphic groups, i.e. such that the action of on has only finitely many orbits for each . These groups are familiar to model theorists as automorphism groups of categorical structures. Todor provides a classification in the case when the underlying structure has trivial algebraic closure, and gives some promising partial results in the general case. In fact, this subject appears to have a lot to do with model theory. I am involved in a project together with Itai Ben Yaacov, of an abstract model theoretic approach to de Finetti’s theory in terms of the forking calculus, canonical bases and Morley sequences in the context of an arbitrary stable firstorder theory, in the sense of continuous logic (which specializes to the classical case considering the theory of valued random variables equipped with the metric).
Also I gave what was probably my last talk as a “student”. I spoke about some new results with Pierre Simon and Anand Pillay concerning definable topological dynamics in NIP theories. The slides are available here. We show that notions like definable (extreme) amenability of a definable group, as well as various model theoretic components, are not affected by adding externally definable sets to the picture (that is, passing to a Shelah’s expansion of a model). These facts appear to have some applications to the questions of Newelski on describing in terms of the socalled Ellis group.
Posted in Conferences, General, Model Theory
Tagged Artem Chernikov, conference, connected component, de Finetti, Forking, Luminy, MALOA, NIP, stability, VC density
“Model theory and applications to geometry”, Satellite workshop associated with LC 2013, Lisbon, 18th19th of July 2013
The aim of the workshop is to give the opportunity to the people attending Logic Colloquium 2013 and to all the people interested, to hear more about the recent developments in model theory and its applications and connections to real geometry.
List of confirmed speakers:
Raf Cluckers (Université de Lille and KULeuven)
Georges Comte (Université de Savoie – Chambéry)
Antongiulio Fornasiero (Seconda Università di Napoli)
Isaac Goldbring (UIC)
François Loeser (Université Pierre et Marie Curie)
Chris Miller (Ohio State University)
Giovanni Morando (Università di Padova)
JeanPhilippe Rolin (Université de Bourgogne)
You can find information about travelling and accommodation on the webpage of the workshop: http://ptmat.fc.ul.pt/~tservi/Workshop%202013/
REGISTRATION: if you are planning on attending this event, please contact tamara.servi@gmail.com
We hope to see you in Lisbon!
The Model Theory Group at CMAF
Universidade de Lisboa
Posted in Announcements, Conferences, Model Theory
Tagged Applications of Model Theory, conference, Lisbon, Model theory
Postdoc in model theory (Caserta, Italy)
I think that we might use this platform to spread information about upcoming meetings and position openings. The following is a message from Paola D’Aquino (Paola.DAQUINO@unina2.it)
Dear All,
there is a two year postdoc position in model theory in my department in Caserta. It will be advertised by end of April and the deadline for applying is around end of May. It’s not required to have a Ph.D. by the
deadline for application. The salary is 25,000 per year before taxes. If you know of anyone who can be interested they can write to me for more information.
Best wishes,
Paola
Some counterexamples for forking, dividing, invariance
At the end of my paper with Itay Kaplan “Forking and dividing in NTP2 theories” we had asked several questions, admittedly without giving them much thought. Since 2008 when the paper went in circulation, some people had actually shown interest in those questions. By now two of them are known to have negative answers, one due to Gabriel Conant and one by myself, with very easy examples. I’d like to have them written down for a reference somewhere, so I’ve thought this might be an appropriate place.
Question 1. (rephrased as more elaborate latex is not available here): Is it true that in a simple theory, every type has a global Lascarinvariant extension.
I recall that a complete global type is Lascarinvariant over a small set if whenever and has the same Lascar strong type over as , then . Having the same Lascar strong type means that is equivalent to with respect to every equivalence relation with boundedly many classes which is invariant.
This property is true in the random graph, for example – any type can be extended to a global one without adding any new edges. This is also true in any extensible NIP theory, say in any stalbe theory, any minimal theory (e.g. real closed fields) or any minimal theory (e.g. algebraically closed valued fields), as well as in any theory with definable Skolem functions (e.g. adics). A theory is extensible if every type does not fork over its domain. However, the crucial point is that this property need not be preserved in reducts of the theory, which immediately gives an easy simple counterexample.
Example. Let be the reduct of the random graph given by the ternary relation which holds if and only if and the number of edges between vertices in the set is odd.
Claim.
 is supersimple of rank . Thus, Lascar strong type is determined by the strong type.
 For any set , .
 All pairs of different elements have the same type over .
 Let and . Then is not Lascarinvariant over .
Proof: (1) is because is definable on the set of singletons in the random graph, which is supersimple of rank 1. Now it is a wellknown fact that Lascar strong type is determined by the strong type in supersimple theories.
(2) is easy to see, and (3) is by backandforth.
(4) Assume is Lascarinvariant over , thus invariant over it by (1) and (2). Let , then by (3) either for all or for all . In the first case, let satisfy . Then it is easy to see that — a contradiction. In the other case, take satisfying and check that — a contradiction again.
Thus, by (4) the unique type over the empty set has no global Lascarinvariant extension.
There are various modifications of the question which still make sense, and also one can ask if this property holds in particular algebraic structures of interest. I have some things to say about it, but not this time.
Question 2. “Can similar results be proved for NSOP theories?”
Here “similar results” refers to the main result of the paper, that is that in an NTP2 theory a formula divides over an extension base if and only if it forks over it. Now, Gabe shows in “Forking and dividing in Henson graphs” that it is not the case for the trianglefree random graph. From my own experience, trianglefree random graph seems to demonstrate the failure of all the phenomena which holds for NTP2 theories.
Example. Let be the theory of the trianglefree random graph, and let . Let .
Claim.
 divides over for any .
 does not divide over .
 is an extension base.
 is but .
 However, forking and dividing are the same for complete types.
See Gabe’s article for details and for the general case of Henson graphs.
Still the following part of the question remains open:
Problem.
 Is forking=dividing for complete types?
 Is forking equal to dividing for formulas in NTP1 over models?
Posted in General, Model Theory
Tagged Artem Chernikov, dividing, Forking, Henson graphs, invariant types, NTP1, NTP2, open question, random graph, simple theories, SOPn
On the SOPn hierarchy inside NTP2
In this post I would like to popularize a certain question around Shelah’s classification theory of unstable (and even nonsimple) theories.
1. Tree properties of the first and second kind, and
As usual we are in a very big and saturated monster model of a complete firstorder theory .
Definition. A formula has if there is a tree of tuples and such that:
• is consistent for any
• is inconsistent for any mutually incomparable .
Otherwise we say that is , and is if every formula is.
Definition. A formula has if there is an array of tuples such that is inconsistent for every and is consistent for any . Otherwise we say that is , and is if every formula is.
It is an easy exercise to see that each of this properties implies the usual tree property, that is the failure of simplicity. An important insight of Shelah is that failure of simplicity always occurs in one of these two explicit ways.
Theorem. is simple if and only if it is both and .
These tree properties were introduced and studied by Shelah in [1] and [2]. reappeared on the scene after it arose in the work on forking and dividing in NIP theories in [3] and was studied further in [4] and [5]. reappears in [6]. I will probably return to these in future postings.
2. On the hierarchy inside
We recall the definition of for from [7,Definition 2.5], another hierarchy introduced by Shelah in order to study nonsimple theories without the strict order property:
Definition. 1. Let . A formula has if there are tuples such that:
 There is an infinite chain: for all
 There are no cycles of length : .
2. has if and only if it has .
3. .
4. By Shelah’s theorem we see that restricting to theories, the last 2 items coincide.
The following are standard examples showing that the hierarchy is strict for :
Example. [7, Claim 2.8]
1. The usual example of a theory which is notsimple and is the model companion of the theory of a parametrized family of equivalence relations with infinitely many infinite classes (see [8]).
2. For , let be the model completion of the theory of directed graphs (no selfloops or multiple edges) with no directed cycles of length . Then it has but not .
3. For odd , the model completion of the theory of graphs with no odd cycles of length , has but not .
4. Consider the model companion of a theory in the language saying:
 ,
 ,
 ,
 if then .
It eliminates quantifiers.
However, all these examples have .
Proof:
(1) It is immediate that the formula has .
(2) Let . For we choose sequencese such that and for all , and these are the only edges around — it is possible as no directed cycles are created. Now for any , if there is , then we would have a directed cycle of length — a contradiction. On the other hand, given and there has to be an element as there are no directed cycles created. Thus has .
(3) and (4) Similar.
This naturally leads to the following question (which I originally asked in [4]):
Question: Is the hierarchy strict for theories? Note that the strictness of the implication is open even in general.
Even an example of a nonsimple theory with and is missing. In [2, §7, Exercise 7.12] Shelah suggests an example of such a theory as an exercise. However, I couldn’t make sense of the suggested example. Perhaps someone else would?
References
 Saharon Shelah, “Simple unstable theories”, Ann. Math. Logic, http://dx.doi.org/10.1016/00034843(80)900091
 Saharon Shelah, “Classification theory and the number of nonisomorphic models”, NorthHolland Publishing Co., 1990
 Artem Chernikov and Itay Kaplan, “Forking and dividing in NTP2 theories”, J. Symbolic Logic, 2012
 Artem Chernikov, “Theories without the tree property of the second kind”, http://arxiv.org/abs/1204.0832
 Itai Ben Yaacov and Artem Chernikov, “An independence theorem for NTP2 theories”, arXiv:1207.0289
 Byunghan Kim and HyeungJoon Kim, “Notions around tree property 1”, Annals of Pure and Applied Logic, 2011
 Saharon Shelah, “Toward classifying unstable theories”, Ann. Pure Appl. Logic, 1996
 Saharon Shelah and Alex Usvyatsov, “More on SOP1 and SOP2”, arXiv:math/0404178
Posted in General, Model Theory
Tagged Artem Chernikov, classification theory, NTP1, NTP2, open question, simple theories, SOPn, tree property