Problems in Extremal Set Theory

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The PI and his coauthors will work on extremal problems for set systems using combinatorial, algebraic, and probabilistic methods. The basic problem of extremal set theory is to determine the maximum number of subsets a finite set can have without containing some fixed forbidden configuration. Many special cases are famous problems that have been open for over 50 years. Nevertheless, substantial work on closely related issues has occurred recently and the PI is part of these projects. The PI will focus on three major questions:the Turan conjecture for the complete 3-graph on four points, Chvatals conjecture on simplices, and sharpening the Frankl-Rodl omitted intersection theorem.The general topic of finite set systems has connections to diverse areas of mathematics (combinatorial geometry, design theory, partially ordered sets, additive number theory), and also to academic disciplines with concrete applications in everyday life (coding theory, information theory, optimization and scheduling problems, computer science). As an example, modern communication would be impossible without the existence of codes, or objects designed to relay information faithfully even in the face of distortion.Extremal problems for set systems play an important role in constructing codes for various situations.