Additive Combinatorics and Ramsey theory

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This proposal contains central problems in additive combinatorics and Ramsey theory. Additive combinatorics is at the intersection of number theory and discrete mathematics, and Ramsey theory addresses the existence of order in all sufficiently large structures. Previous progress on these and related problems by the PI and his collaborators used powerful techniques from diverse areas of mathematics, including from combinatorics, probability, analytic number theory, algebraic geometry, model theory, and topology. The particular problems chosen and the techniques that have just begun to be explored appear ripe for more substantial progress. Graduate students will be trained during this award.The first area in this proposal concerns subset sums. In this area, the PI and his collaborators have recently solved some longstanding open problems using new techniques. However, major problems remain open. For example, the PI plans to work on better estimating the size of the largest non-averaging subset of the first n positive integers. Another example the PI plans to work on is the Erdős distinct subset sums conjecture. Partial progress on these problems by the PI and his collaborators suggest further substantial progress could be within reach. The second area is on estimating Ramsey numbers. The PI will work on proving new bounds for classical graph and hypergraph Ramsey numbers as well as more recent problems on sparse directed graphs. The PI has made progress using novel techniques, and further development of these methods are expected to yield substantial progress on central questions in this area.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.