- Titel
- Problems in Extremal and Probabilistic Combinatorics
- Zitierfähige Url:
- https://nbn-resolving.org/urn:nbn:de:bsz:15-qucosa2-1061124
- Datum der Einreichung
- 01.04.2026
- Datum der Verteidigung
- 15.07.2026
- Abstract (EN)
- This thesis presents advances in two central areas of modern combinatorics: Extremal Hypergraph Theory and Random Graph Theory. The results are presented in three parts. First, we resolve a longstanding open problem in the theory of random graphs known as the Kim-Vu Sandwich Conjecture (2004). This conjecture posits that a random $d$-regular graph $G_{n,d}$ can be coupled between two binomial random graphs $G(n, (1-\eps)d/n)$ and $G(n, (1+\eps)d/n)$ with high probability. Second, we investigate the classical Tur\'{a}n problem for $r$-uniform hypergraphs. Motivated by the search for a higher-dimensional analogue of Mantel's Theorem, we study when the Tur\'{a}n density of $r$-uniform hypergraphs is exactly $r!/r^r$. Finally, we explore the concept of uniform Tur\'{a}n density, a variation of the extremal problem that imposes density constraints on all linear-sized induced subgraphs. We expand the set of known uniform Tur\'{a}n densities by proving the existence of a family of 3-uniform hypergraphs with uniform Tur\'{a}n density exactly equal to $8/27$.
- Andere Ausgabe
- A proof of the Kim-Vu sandwich conjecture
Link: https://arxiv.org/abs/2510.20765 - An improved hypergraph Mantel's Theorem
Link: https://arxiv.org/abs/2503.14474 - Hypergraphs with uniform Turán density equal to 8/27
Link: https://arxiv.org/abs/2407.05829 - Freie Schlagwörter (EN)
- combinatorics, random graphs, Turán problems
- Klassifikation (DDC)
- 500
- Den akademischen Grad verleihende / prüfende Institution
- Universität Leipzig, Leipzig
- Version / Begutachtungsstatus
- publizierte Version / Verlagsversion
- URN Qucosa
- urn:nbn:de:bsz:15-qucosa2-1061124
- Veröffentlichungsdatum Qucosa
- 21.07.2026
- Dokumenttyp
- Dissertation
- Sprache des Dokumentes
- Englisch
- Lizenz / Rechtehinweis
CC BY-NC 4.0