The investigator studies how local properties affect the global parameters of different combinatorial structures. This is a very general framework of the so called Turan number problems. The investigator emphasizes four different aspects 1. To study the Turan numbers of triple systems and multigraphs, as a tool to achieve a general theory for hypergraphs. 2. To study covering radius problems, especially concerning constant weight codes where Turan numbers naturally emerge. 3. To study more general coding theory problems, like superimposed codes, identifying codes which lead to hypergraph intersection problems. 4. To find geometrical, and algebraic representations, like Lovasz' Shannon capacity bound, Ramanujan graphs, polarity graphs, Prague dimension of graphs, where Turan numbers naturally emerge.
Most finite problems can be formulated as extremal graph or hypergraph problems. Extremal combinatorics applies a broad array of tools and results from other fields of mathematics like number theory, linear and commutative algebra, probability theory, geometry, and information theory. On the other hand it has a number of interesting applications in all parts of combinatorics, and in geometry, integer programming, computer science, coding theory, dimension theory of partially ordered sets, encryptions. Applications of extremal combinatorics and coding theory in computer science and communication theory are indispensable.
