Artem Chernikov

Topics in Model Theory and Combinatorics, MATH818J (Spring 2026, UMD)

• In this course we cover
  
  
• 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.
  
  
• 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  Hans Adler, "Strong theories, burden, and weight", unpublished [link]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.
• 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.
  
  
• 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.
• 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
  
  
• 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.
• 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.
  
  
• 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.
• Lec 10 (Mar 5) Finishing existence of L_str indiscernibles; TP + NTP2 implies L_str tree witnessing TP
  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.
  
  
• Lec 11 (Mar 10) Starting TP implies TP1 or TP2
  Theorem 4.12 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]
  
• Lec 12 (Mar 12) Finishing TP implies TP1 or TP2
  Theorem 14 in Hans Adler, "Strong theories, burden, and weight", unpublished [link]
  
  
• Lec 13 (Mar 24) Simple theories. Equivalence of NTP and the local character of dividing
  Casanovas, Enrique. "Simplicity simplified." Revista Colombiana de Matemáticas 41 (2007): 263-277.
  Artem Chernikov, "Lecture notes on stability theory", Section 3.5
  
• Lec 14 (Mar 26) Further properties of dividng. Kim's lemma
  Casanovas, Enrique. "Simplicity simplified." Revista Colombiana de Matemáticas 41 (2007): 263-277.
  Artem Chernikov, "Lecture notes on stability theory", Sections 3.4,3.5
  
  
• Lec 15 (Apr 2) Every set is an extension base. Existence of Morley sequences. Forking = dividing. Ternary independence relations
  Artem Chernikov, "Lecture notes on stability theory", Section 3.5
  
  
• Lec 16 (Apr 7) Properties of forking in arbitrary and in simple theories (symmetry, transitivity, etc.). Bounded relations.
  Casanovas, Enrique. "Simplicity simplified." Revista Colombiana de Matemáticas 41 (2007): 263-277.
  Artem Chernikov, "Lecture notes on stability theory", Sections 3.5, Proposition 3.7.12
  
• Lec 17 (Apr 9) Lascar strong types, thick formulas, strong automorphisms
  Casanovas, Enrique. "Simplicity simplified." Revista Colombiana de Matemáticas 41 (2007): 263-277.
  
  
• Lec 18 (Apr 14) Towards the proof of the independence theorem: the chain condition for forking and auxiliary lemmas
  Section 6 of Casanovas, Enrique. "Simplicity simplified." Revista Colombiana de Matemáticas 41 (2007): 263-277.
  
• Lec 19 (Apr 16) Proof of the independence theorem. Equality of Lascar strong types is type-definable
  Section 6 of Casanovas, Enrique. "Simplicity simplified." Revista Colombiana de Matemáticas 41 (2007): 263-277.
  
  
• Lec 20 (Apr 21) Finite satisfiability, Lascar invariance, and some properties of forking in NIP theories
  Chernikov, Artem and Kaplan, Itay. "Forking and dividing in NTP2 theories", Journal of Symbolic Logic 77 (1), 1-20 (2012)
  Adler, Hans. "An introduction to theories without the independence property." Archive for Mathematical Logic 5 (2008)
  
• Lec 21 (Apr 23) Forking and dividing in NTP2 theories. Dividing over a model is always witnessed by a coheir sequence.
  Chernikov, Artem and Kaplan, Itay. "Forking and dividing in NTP2 theories", Journal of Symbolic Logic 77 (1), 1-20 (2012)
  Adler, Hans "Kim’s lemma for NTP2 theories: a simpler proof of a result by Chernikov and Kaplan." Rend. Semin. Mat. Univ. Politec. Torino 72.3-4 (2014): 121-126
  
  
• Lec 22 (Apr 28) Vacuum cleaner lemma and the existence of strict invariant types over models in NTP2
  Chernikov, Artem and Kaplan, Itay. "Forking and dividing in NTP2 theories", Journal of Symbolic Logic 77 (1), 1-20 (2012)
  Adler, Hans "Kim’s lemma for NTP2 theories: a simpler proof of a result by Chernikov and Kaplan." Rend. Semin. Mat. Univ. Politec. Torino 72.3-4 (2014): 121-126.
  
• Lec 23 (Apr 30) Strict Morley sequences witness dividing, and forking equals dividing over models in NTP2
  Chernikov, Artem and Kaplan, Itay. "Forking and dividing in NTP2 theories", Journal of Symbolic Logic 77 (1), 1-20 (2012)
  Adler, Hans "Kim’s lemma for NTP2 theories: a simpler proof of a result by Chernikov and Kaplan." Rend. Semin. Mat. Univ. Politec. Torino 72.3-4 (2014): 121-126.
  
  
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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]