Model theory of pseudofinite structures
This is the teaching webpage for the online seminar coordinated with Itay Kaplan on Model theory of pseudofinite structures and its applications to combinatorics.
Time: every Wednesday 16:00-17:45 Israel summer time (GMT+3) from 25 March, 2020 to June 24, 2020.
Slides and video for 25/03/2020.
Slides and video for 01/04/2020.
Slides and video for 22/04/2020.
Slides from Daniel Losub for 06/05/2020.
Slides and note of proof for 13/05/2020.
Slides and note of proof for 20/05/2020.
Slides and note of proof for 27/05/2020 and note of Gowers Lemma from Weikung He.
Slides from Ori Segel for 03/06/2020.
Slides for 10/06/2020.
Slides and note of proof for 17/06/2020.
Topics and references:
Counting dimension:
E. Hrushovski, On pseudo-finite dimensions, Notre Dame J. Form. Log. 54(3-4) (2013) 463–495. (Section 2)
Artem Chernikov, note on his course: Pseudofinite model theory, https://www.math.ucla.edu/~chernikov/teaching/19F-MATH223M/Notes.pdf
Erdös-Hajnal for stable graphs:
Main reference: A. Chernikov and S. Starchenko, A note on the Erdős-Hajnal property for stable graphs, Proc. Am. Math. Soc. 146.2 (2018): 785--790. arXiv: https://arxiv.org/abs/1504.08252
Further reference: Maria Chudnovsky, The Erdös-Hajnal conjecture -- A survey, J. Graph Theory, 75(2) (2014) : 178--190. https://web.math.princeton.edu/~mchudnov/EHsurvey.pdf
Larsen-Pink inequality:
Main reference: E. Hrushovski and F. Wagner, Counting and dimensions, in Model theory with applications to algebras and analysis, volume 350 of London Mathematical Society Lecture Notes Series, (Cambridge University Press, 2008,) pp.~161--176.
http://www.logique.jussieu.fr/modnet/Publications/Preprint%20server/papers/5/5.pdf
Further reference: E. Hrushovski, Stable group theory and approximate subgroups, J. Am. Math. Soc. 25(1) (2012) 189--243. (Section 5.4-Lemma 5.8)
Elekes-Szabó:
Main reference: M. Bays and E. Breuillard, Projective geometries arising from Elekes-Szabó problems, arXiv: https://arxiv.org/abs/1806.03422 (Section 2 and section 3)
Further reference: G.Elekes and E. Szabó, How to find groups? (and how to use them in Erdös geometry?), Combinatorica, 32 (2012): 537--571.
Product-free sets:
Main reference: D. Palacín, On compactifications and product-free sets, J. London Math. Soc. (2) 101 (2020): 145--174. arXiv: https://arxiv.org/pdf/1807.10113.pdf
Further reference: E. Hrushovski, Stable group theory and approximate subgroups, J. Am. Math. Soc. 25(1) (2012) 189--243. (Section 1,2,3)