This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
The PI has studied how local properties affect the global parameters of various combinatorial structures. This is a very general framework of the so-called Turan number problems. The PI plans to continue his work on this topic and investigates four different aspects: 1. To study the Turan numbers of triple systems and multigraphs, as a tool to achieve a general theory for r-graphs, e.g., to prove Kalai's conjecture. 2. To investigate natural generalizations of Turan's question, like the number of substructures, stability questions, and consider other host-graphs, like the hypercube. 3. To study general coding theory, design-theory, combinatorial geometry problems, geometric and algebraic graph representations, which lead to hypergraph intersection and other Turan type problems, e.g., superimposed and covering codes, and the completion problem of partial G-designs. 4. To find geometric/algebraic graph representations where Turan numbers naturally emerge, e.g., Prague-dimension, intersection and geometric representations of graphs.
The subject of this proposal is the effect of local properties on global parameters of combinatorial structures, in other words, extremal combinatorics. The PI continue to find applications in theoretical computer science, coding theory and discrete geometry. Combinatorics deals with finite but very large problems arising from computer science, data mining, and communications. Extremal combinatorics applies a broad array of tools and results from other fields of mathematics, on the other hand, it has a number of interesting applications in in geometry, integer programming, computer science, coding theory, dimension theory of partially ordered sets, and cryptography. Combinatorics is the theoretical basis of the economical, fast and reliable algorithms to store and reach data structures. Applications of extremal combinatorics and coding theory in computer science, computer graphics and in communication theory are indispensable.
