Topics in Discrete mathematics/Number theory

Christian Elsholtz
Institut für Analysis und Zahlentheorie
Sommersemester 2021

About every 2 weeks we have an exercise class, online

The exercise sheets are in the teach center.

The lecture-videos will be pre-recorded and are hopefully available by the time of the course times. They are available via TUbe (in the teach center). We will have some online sessions, e.g. for asking about material.
Questions that are of general interest (such as homework or course material) can be asked in the Teach Center, or live sessions.
In the Teach Center I will often have relevant course material, such as riginal papers, sometimes own notes.
Some topics:
1) Construction of progression free sets
(Erdos-Turan, Salem-Spencer, Behrend, Elkin, Varnavides)
2) Van der Waerden's theorem, Hales-Jewett theorem, caps (Croot-Lev-Pach, Ellenberg-Gijswijt, Elsholtz-Pach), Kakey.
3) Combinatorial Nullstellensatz, Polynomial method for sumsets, Cauchy-Davenport, EGZ, Erdos-Heilbronn, zero-sum problems, Reiher, Elsholtz
Further topics can include
arithmetic progressions in sparse sets, Folkman's theorem
Hilbert cubes

(less likely:) - Ramsey theory
- Universal sequences (On strings containing all subsets as substrings Original, Discrete Mathematics, Volume 21, Issue 3, 1978, Pages 253-259 Witold Lipski Jr.)
- Sperner's Lemma (following Jukna's book)
LYM inequality
(Paper by Fox and Lovasz on regularity lemma (arXiv:1606.01230))
- Varnavides in integers and in vector spaces.
- Thue-Morse sequence, almost periodic sequences
(following Jacobs Einf├╝hrung in die Kombinatorik)