Hypergraphs, Ramsey Theory and Extremal Combinatorics

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The proposed research concentrates mainly on problems in Ramsey theory and on extremal graph and hypergraph theory. Both of these areas belong to the mainstream of contemporary discrete mathematics and are often motivated by applications in other areas, such as theoretical computer science. In tackling these problems, a variety of combinatorial methods will be used, such as probabilistic methods and applications of regularity techniques. The PI also plans to develop new techniques leading to the construction of graphs with certain special properties. The Ramsey type problems suggested in the proposal include the investigation of the asymptotic behavior of hypergraph Ramsey numbers, of induced Ramsey numbers, of size-Ramsey numbers, as well of the Erdos-Rogers function. The PI also intends to study some problems in structural Ramsey theory. The extremal problems that the PI intends to investigate include Ramsey-Turán problems and the study of Hamilton cycles and matchings in hypergraphs, as well as extremal problems regarding random subsets of integers.This research will serve as a basis for the mentoring activities of the PI. Concepts and questions of combinatorial character appear naturally in other branches of mathematics and, more broadly speaking, in computer science and in the physical, biological, and social sciences, making combinatorial mathematics of interest not only within mathematics itself, but within the sciences as a whole.