Additive Combinatorics and Ramsey theory

加法组合学和拉姆齐理论

• 批准号: 2154129
• 负责人: Jacob Fox
• 金额: $ 30万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2154129&HistoricalAwards=false
• 关键词: Additive; Combinatorics; Ramsey; theory

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.

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，而拉姆齐理论解决了所有足够大的结构中序的存在问题。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。所选择的特定问题和刚刚开始探索的技术似乎已经成熟，可以取得更实质性的进展。研究生将在此奖项期间接受培训。这项建议的第一个领域涉及子集总和。在这一领域，PI和他的合作者最近使用新技术解决了一些长期悬而未决的问题。然而，主要问题仍然悬而未决。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ERDőS不同子集和猜想。PI和他的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指日可待的。第二个领域是对拉姆齐数字的估计。PI将致力于证明经典图和超图Ramsey数的新界，以及稀疏有向图上的更新问题。PI使用新技术取得了进展，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。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

The upper logarithmic density of monochromatic subset sums

• DOI: 10.1112/mtk.12167
• 发表时间: 2021-05
• 期刊: Mathematika
• 影响因子: 0.8
• 作者: D. Conlon;J. Fox;H. Pham

Quasiplanar graphs, string graphs, and the Erdos-Gallai problem

拟平面图、弦图和 Erdos-Gallai 问题

• 发表时间: 2023
• 期刊: Graph Drawing and Network Visualization. GD 2022. Lecture Notes in Computer Science
• 作者: Fox, J.;Pach, J.;Suk, A.
• 通讯作者: Suk, A.

Off-diagonal book Ramsey numbers

非对角书拉姆齐数

• DOI: 10.1017/s0963548322000360
• 发表时间: 2023
• 期刊: Probability and Computing
• 作者: Conlon, David;Fox, Jacob;Wigderson, Yuval
• 通讯作者: Wigderson, Yuval

Geometric and o-minimal Littlewood–Offord problems

几何和 o 最小 Littlewood–Offford 问题

• DOI: 10.1214/22-aop1590
• 发表时间: 2023
• 期刊: The Annals of Probability
• 作者: Fox, Jacob;Kwan, Matthew;Spink, Hunter
• 通讯作者: Spink, Hunter

Which graphs can be counted in $C_4$-free graphs?

哪些图可以算在无 $C_4$ 图中？

• DOI: 10.4310/pamq.2022.v18.n6.a4
• 发表时间: 2022
• 期刊: Pure and Applied Mathematics Quarterly
• 影响因子: 0.7
• 作者: Conlon, David;Fox, Jacob;Sudakov, Benny;Zhao, Yufei
• 通讯作者: Zhao, Yufei