2018 – 2022
- Extremal Combinatorics, Extremal Set Theory, Graph Theory
- Advisor: Ervin Győri
salianika@gmail.com
Mathematics is the science that lays the groundwork for all other sciences. Combinatorics and Graph Theory are fields of mathematics that have played a similar role within mathematics. They emerged as independent disciplines with the rise of computer science and its need for discrete structures and found applications in various branches of mathematics. Extremal combinatorics is a sub-field of combinatorics that explores how local properties affect global properties of combinatorial structures. These questions are not only beautiful in their own right but also find applications in number theory, discrete geometry, and theoretical computer science, to name a few.
A typical problem in extremal combinatorics is to find the largest or the smallest size of a combinatorial object that satisfies certain constraints. For example, how many edges can a graph have if it does not contain a certain subgraph? This is a so-called forbidden subgraph problem (also referred to as the Turán problem), one of the most classic and important problems in the area. Addressing it for the forbidden 4-cycle C_4, a remarkable theorem of Erdős says that the maximum number of edges in a C_4-free graph is of order n^{3/2}, where n is the number of vertices. Erdős used this theorem in order to prove an asymptotic result for a famous number theory problem, namely the maximum number of bounded integers with distinct products.
Extremal combinatorics is an active and rich field, with many open problems and recent breakthroughs. Some of the current research directions include Ramsey theory, Turán type problems including generalized Turán problems, stability and saturation problems for discrete structures, extremal problems for hypergraphs and set systems, for graphs with a bounded degree or girth, for graphs with forbidden induced subgraphs, and for graphs with algebraic or geometric structure. The goal of this project is to pursue further research in these areas and address a number of long-standing open problems, with a particular focus on Turán numbers of bipartite graphs, extremal problems in geometric settings, and Turán problems for hypergraphs.
le23@nyu.edu
Research Associate, New York University Abu Dhabi, UAE
A. Razmadze Mathematical Institute.
Budapest, Hungary
Tbilisi, Georgia
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