Additive Combinatorics and Ramsey theory

这个建议包含了加法组合学和拉姆齐理论中的核心问题。加法组合学是数论和离散数学的交叉点，拉姆齐理论解决了所有足够大的结构中存在的顺序。PI和他的合作者以前在这些问题和相关问题上的进展使用了来自不同数学领域的强大技术，包括组合学，概率论，解析数论，代数几何，模型论和拓扑学。所选择的具体问题和刚刚开始探索的技术似乎已经成熟，可以取得更大的进展。研究生将在这个奖项的培训。在这个建议的第一个领域涉及子集和。在这个领域，PI和他的合作者最近使用新技术解决了一些长期存在的开放问题。然而，主要问题仍然存在。例如，PI计划更好地估计前n个正整数的最大非平均子集的大小。PI计划研究的另一个例子是Erdos的不同子集和猜想。PI及其合作者在这些问题上取得的部分进展表明，进一步的实质性进展可能是触手可及的。第二个领域是关于Ramsey数的估计。PI将致力于证明经典图和超图Ramsey数的新界限，以及稀疏有向图的最新问题。 PI已经使用新的技术取得了进展，这些方法的进一步发展有望在这一领域的核心问题上取得实质性进展。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.