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Course, academic year 2023/2024
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Infinite sets - NMAI074
Title: Nekonečné množiny
Guaranteed by: Department of Applied Mathematics (32-KAM)
Faculty: Faculty of Mathematics and Physics
Actual: from 2022
Semester: winter
E-Credits: 3
Hours per week, examination: winter s.:2/0, Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: taught
Language: Czech
Teaching methods: full-time
Teaching methods: full-time
Additional information: https://kam.mff.cuni.cz/~kyncl/infinite
Guarantor: doc. Mgr. Jan Kynčl, Ph.D.
Class: DS, diskrétní modely a algoritmy
Informatika Bc.
Informatika Mgr. - Diskrétní modely a algoritmy
Classification: Informatics > Discrete Mathematics
Annotation -
This course is a sequel to Set theory (NAIL063). We will focus mostly on combinatorial properties of infinite sets and graphs. We will also see examples of "elementary" combinatorial statements whose validity depends on the chosen axioms. It is assumed that the students have basic knowledge of set theory (NAIL063), for some applications basics of group theory and measure theory would also be helpful.
Last update: Kynčl Jan, doc. Mgr., Ph.D. (09.05.2018)
Course completion requirements -

Oral exam.

Last update: Kynčl Jan, doc. Mgr., Ph.D. (29.05.2019)
Literature -

B. Balcar, P. Štěpánek, Teorie množin, Academia, Praha, 2001

K. Hrbacek, T. Jech, Introduction to Set Theory, 3.ed., Marcel Dekker, 1999

T. Jech, Set theory, Springer, 2003

B. Bollobas, Modern Graph Theory, Graduate Texts in Mathematics 184, Springer-Verlag, New York, 1998

R. Diestel, Graph theory, Fifth edition, Graduate Texts in Mathematics, 173, Springer, Berlin, 2017

R. Graham, B. Rothschild, J. Spencer, Ramsey theory, Second edition, Wiley-Interscience Series in Discrete Mathematics and Optimization, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1990.

H. J. Prömel, Ramsey theory for discrete structures, With a foreword by Angelika Steger, Springer, Cham, 2013

Last update: Kynčl Jan, doc. Mgr., Ph.D. (09.05.2018)
Requirements to the exam -

The exam will be oral based on the recommended literature and the material that was presented. The exam can also be in a distance form.

Last update: Kynčl Jan, doc. Mgr., Ph.D. (10.10.2020)
Syllabus -
  • Ordinal type of a well-ordered set, transfinite recursion, Zorn's lemma
  • Ordinal arithmetics, Goodstein sequences
  • Cardinal numbers and cardinal arithmetics
  • Infinite Ramsey-type theorems
  • Infinite graphs
  • Applications of the axiom of choice in particular in combinatorics and geometry

Last update: Kynčl Jan, doc. Mgr., Ph.D. (06.10.2019)
 
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