FRG : Collaborative Research : Pseudorandomness in Ramsey Theory

拉姆齐理论指的是数学中一大堆深刻的结果，它们有一个共同的主题：在大型组合结构中找到统一的子结构。它现在是现代组合学中最核心的领域之一。这门学科是由弗兰克·拉姆齐于1930年创立的，当时他正在研究逻辑系统的可判断性，他的基本结果现在被称为拉姆齐定理。1935年，保罗·埃尔多斯和乔治·塞克尔斯在研究一个看似无关的几何问题时重新发现了他的定理。考虑到这些不同的起源，Ramsey定理在其他数学领域有广泛的应用也就不足为奇了，包括逻辑、几何、数论和理论计算机科学。这个专注的研究小组的目标是获得经典Ramsey数的新界。该小组将在该领域使用广泛的工具和技术，包括概率方法、步进引理和伪随机图理论。最近，Mubayi和Verstraete在Alon和Rodl的工作基础上建立了Ramsey数和伪随机图之间令人惊讶的联系，从而将该领域的重点从随机图转移到伪随机图。此外，最近在超图Ramsey数方面取得了实质性的进展，我们现在知道其中许多数字的塔增长率。预计在这些问题上的进一步工作也将导致新的方法和应用。最后，将培训和支持相当数量的学生和早期职业研究人员，研究产生的合作成果将在会议、研讨会和出版物上广泛传播。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

Ramsey theory refers to a large body of deep results in mathematics which have a common theme: find uniform substructures in large combinatorial structures. It is now one of the most central areas in modern combinatorics. The subject was founded by Frank Ramsey in 1930 while studying the decidability of logical systems and his foundational result is now known as Ramsey’s theorem. His theorem was rediscovered in 1935 by Paul Erdos and George Szekeres while studying a seemingly unrelated geometric question. Given these diverse origins, it is not surprising that Ramsey’s theorem has had a wide range of applications in other areas of mathematics including logic, geometry, number theory, and theoretical computer science.The goal of this focused research group is to obtain new bounds for classical Ramsey numbers. The group will use a wide range of tools and techniques in the area including the probabilistic method, the stepping-up lemma, and the theory of pseudorandom graphs. Very recently, Mubayi and Verstraete established a surprising connection between the Ramsey numbers and pseudorandom graphs based on the work of Alon and Rodl, thus moving the emphasis of the field from random graphs to pseudorandom graphs. Moreover, substantial progress has recently been made on hypergraph Ramsey numbers, where we now know the tower growth rate for many of these numbers. It is expected that further work on these problems will lead to new methods and applications as well. Finally, a substantial number of students and early-career researchers will be trained and supported, and the collaborative results arising from the research will be disseminated widely at conferences, workshops and via publications.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.