Topics in Model Theory and Combinatorics, MATH818J (Spring 2026, UMD)
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In this course we cover
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Lecture 1 (Feb 3) - Classification of first-order theories, tree properties TP, TP1, TP2
A (partial) map of the classification-theoretic universe, by Gabe Conant [link]
See the intro in "On model-theoretic tree properties", Artem Chernikov, Nicholas Ramsey, Journal of Mathematical Logic, 16(2), (2016) and references there; or these slides for a brief summary.
- Lec 2 (Feb 5) Indiscernible (sub)sequences; Erdős-Rado; mutual indiscernibility
Sections 1,2 and references in Chernikov, Artem. "Theories without the tree property of the second kind" Annals of Pure and Applied Logic 165.2 (2014): 695-723.
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Lec 3 (Feb 10) (Strongly) indiscernible arrays; burden
Hans Adler, "Strong theories, burden, and weight", unpublished [link]
Sections 1,2 of Chernikov, Artem. "Theories without the tree property of the second kind" Annals of Pure and Applied Logic 165.2 (2014): 695-723.
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Lec 4 (Feb 12) A formula-free characterization of NTP2 using strongly indiscernible arrays
Section 2 of Chernikov, Artem. "Theories without the tree property of the second kind" Annals of Pure and Applied Logic 165.2 (2014): 695-723.
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Lec 5 (Feb 17) Submultiplicativity of "burden+1" and reduction of TP2 to one variable formulas
Sections 2,3 of Chernikov, Artem. "Theories without the tree property of the second kind" Annals of Pure and Applied Logic 165.2 (2014): 695-723.
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Lec 6 (Feb 19) Generalized indiscernibles and structural Ramsey theory
Scow, Lynn. "Characterization of NIP theories by ordered graph-indiscernibles." Annals of Pure and Applied Logic 163.11 (2012): 1624-1641
Scow, Lynn. "Indiscernibles, EM-types, and Ramsey classes of trees." (2015): 429-447
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Lec 7 (Feb 24) More on generalized indiscernibles, structural Ramsey and Fraïssé limits, two kinds of tree-indexed indiscernibles (L_s and L_str)
Section 2 of "On model-theoretic tree properties", Artem Chernikov, Nicholas Ramsey, Journal of Mathematical Logic, 16(2), (2016)
Kim, Byunghan, Hyeung-Joon Kim, and Lynn Scow. "Tree indiscernibilities, revisited." Archive for Mathematical Logic 53.1 (2014): 211-232.
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Lec 8 (Feb 26) Existence of L_s indiscernibles (i.e. with predicates for the levels)
Theorem 4.3 in Kim, Byunghan, Hyeung-Joon Kim, and Lynn Scow. "Tree indiscernibilities, revisited." Archive for Mathematical Logic 53.1 (2014): 211-232.
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Lec 9 (Mar 3) Finishing existence of L_s indiscernibles; existence of L_str indiscernibles (i.e. only with level comparison relation)
Theorem 4.12 in Kim, Byunghan, Hyeung-Joon Kim, and Lynn Scow. "Tree indiscernibilities, revisited." Archive for Mathematical Logic 53.1 (2014): 211-232.
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Lec 10 (Mar 5) Finishing existence of L_str indiscernibles; TP implies TP1 or TP2
Theorems 4.12 and 5.9 in Kim, Byunghan, Hyeung-Joon Kim, and Lynn Scow. "Tree indiscernibilities, revisited." Archive for Mathematical Logic 53.1 (2014): 211-232.
Theorem 14 in Hans Adler, "Strong theories, burden, and weight", unpublished [link]
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Some further references
Tent, Katrin, and Martin Ziegler. A course in model theory. No. 40. Cambridge University Press, 2012.
Marker, David. Model theory: an introduction. New York, NY: Springer New York, 2002.
Artem Chernikov, "Lecture notes on stability theory", [link]
Nick Ramsey, "Model tree properties", lecture videos [link]