The PI proposes to study a variety of open problems in extremal combinatorics, an area that deals with the asymptotic relations between various parameters of large discrete structures. This area has grown tremendously in the past few decades, in depth and in breadth, and supplied many spectacular results that affected various other areas of mathematics, such as number theory, group theory, probability theory, information theory and theoretical computer science.
A common theme in many of the problems the PI will investigate is the relation between local and global properties of various discrete structures. For example, suppose a graph contains few copies of some fixed graph; what can one infer from this simple local restriction about other more global properties of this object? Perhaps surprisingly, many of the outstanding open problems in extremal combinatorics can be cast in this manner. One such problem is the celebrated hypergraph removal lemma. An important goal of this project is to obtain the first primitive recursive bounds for this fundamental problem. Another family of such problems are motivated by questions in theoretical computer science and ask to develop fast algorithms for deciding if a discrete structure satisfies some predetermined property. The PI will develop a unifying theory for such problems using various combinatorial tools introduced in the past few years. Another set of problems ask to establish that hypergraphs excluding certain types of sub-structures, possess surprisingly good Ramsey-type properties.
The common thread going through the above problems is Szemeredi's regularity lemma, which is one of the most powerful tools in extremal combinatorics. An important goal of this project is to find new applications of the graph and hypergraph versions this lemma, but more importantly, to develop new efficient variants of the lemma, tailored for specific applications, that would allow us to settle the problems discussed in this proposal.
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