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
