TUHH / Institut für Mathematik / Forschungsgebiete / Research Area: Extremal Set Theory

Research Area: Extremal Set Theory

Working Groups: Lehrstuhl Diskrete Mathematik

Set theory is the mathematical theory of sets. Informally speaking, sets are collections of objects, called elements. A collection of sets is refered to as a family.

Extremal set theory deals with questions such as how many sets can a certain family contain, if all of them have to fulfil certain properties, for example if every set intersects every other set in the family, or no set is a subset of another set. One of the earliest results in this area is the celebrated result by Erdős, Ko, and Rado, which establishes the maximum size of a uniform intersecting family:

Theorem
Let \(k,n \in \mathbb{N}\) with \(2k \leq n\) and let \(\mathcal{F} \subseteq [n]^{(k)}\) be an intersecting family. Then \(|\mathcal{F}| \leq \binom{n-1}{k-1}\) and this bound is sharp.

Literature

[FT2018] P. Frankl and N. Tokushige, Extremal Problems for Finite Sets, Student Mathematical Library, Volume 86, American Mathematical Society, Providence, RI, 2018

Publications and Manuscripts

[1] D. Clemens, S. Das, and T. Tran. Colourings without monochromatic disjoint pairs, European Journal of Combinatorics Volume 70, May 2018, 99-124.

[2] P. Gupta, Y. Mogge, S. Piga, and B. Schülke. \(r\)-cross \(t\)-intersecting families via necessary intersection points, arXiv:2010.11928 (2020)