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Course, academic year 2022/2023
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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
Virtual mobility / capacity: no
State of the course: taught
Language: Czech
Teaching methods: full-time
Additional information:
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 -
Last update: doc. Mgr. Jan Kynčl, Ph.D. (09.05.2018)
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.
Course completion requirements -
Last update: doc. Mgr. Jan Kynčl, Ph.D. (29.05.2019)

Oral exam.

Literature -
Last update: doc. Mgr. Jan Kynčl, Ph.D. (09.05.2018)

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

Requirements to the exam -
Last update: doc. Mgr. Jan Kynčl, Ph.D. (10.10.2020)

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.

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

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