Summer semester 2020

## Discrete Mathematics of Paul Erdős -- NDMI107

• Course description:
Paul Erdős (1913 -- 1996) was an outstanding, prolific, influential, legendary mathematician. We will study a selection of his results in number theory, geometry, Ramsey theory, extremal combinatorial problems, and graph theory that laid the foundations of discrete mathematics before it matured into the rich and vibrant discipline of today. From time to time we will stray from his own work to the work of his confrères and disciples, but we shall never escape the gravitational pull of the great man.

At a leisurely pace, we shall cover a subset of the following topics:
Proof of Bertrand's postulate. Sperner's theorem and the Erdős-Ko-Rado theorem. Turán numbers. Property B and hypergraph colouring. Extremal graph theory. Hamilton cycles. The Friendship Theorem, strongly regular graphs, and Moore graphs of diameter two. Chromatic number of graphs and the probabilistic method. Van der Waerden's theorem and van der Waerden numbers. The Erdős-Szekeres and the Sylvester-Gallai theorems. Delta-systems and Deza's proof of an Erdős-Lovász conjecture. The De Bruijn-Erdős theorem. The Erdős-Rényi random graphs and their evolution. Ramsey's theorem and Ramsey numbers.

 The subject and the instructor, ca.1976
• Instructor: Vašek Chvátal
• Email:     c h v a t a l (at) k a m (dot) m f f (dot) c u n i (dot) c z
• Office hours: by appointment

• Classes: Thursdays 14:00 -- 15:30 in S9

• Prerequisities: Ease with the notation of elementary set theory and with matrix descriptions of systems of linear equations.

### Record of the material covered in class

Date             Material covered
20.2 Proof of Bertrand's postulate up to and including Section 4.
27.2 Proof of Bertrand's postulate concluded.
5.3 Landau's problems. Sperner's theorem and its proof by Sperner.
12.3 Reading assignment: Extremal set theory.
19.3 Reading assignment: Extremal set theory: First four sections (up to page 18).
26.3 Reading assignment: The last section of Extremal set theory and the first two sections of Extremal graph theory.
2.4 Reading assignment: Extremal graph theory.
9.4 Reading assignment: Extremal graph theory.
16.4 Reading assignment: Extremal graph theory.
23.4 Reading assignment: Hamilton cycles.
30.4 Independent study plans.
7.5 Independent study plans.
14.5 Independent study plans.
21.5 Independent study plans.

 Erdős quotations Property is a nuisance. There will be plenty of time to rest in the grave. A theorem a day means promotion and pay; a theorem a year and you're out on your ear. Another roof, another proof. Favourites of Erdős, often attributed to him, but originating elsewhere A mathematician is a machine for turning coffee into theorems. (Alfréd Rényi) Chebyshev said it and I say it again: There is always a prime between n and 2n. (Nathan Fine) Websites dedicated to Erdős Articles on Erdős Two Places at Once (A Remembrance of Paul Erdős) by János Pach Paul Erdős (1913 -- 1996) by László Babai and Joel Spencer The Mathematics of Paul Erdős by László Babai, Carl Pomerance, and Pétér Vértesi Reminiscences of Paul Erdős (1913-1996) by Melvin Henriksen Remembering Paul Erdős by Aleksandar Ivić Biography from MacTutor Obituaries from The New York Times and The Washington Post Two films on Erdős by George Csicsery A book on Erdős Bruce Schechter, My Brain is Open : The Mathematical Journeys of Paul Erdős , Simon & Schuster, 1998.