Probabilistic and Extremal Combinatorics

这个研究项目是对离散数学对象的调查，比如网络和代码。极值组合学专注于更好地理解离散数学对象，优化有趣或理想的属性，而概率组合学研究由一系列随机选择产生的离散数学对象。这两个研究方向密切相关，因为随机算法是构建有趣的离散数学对象的非常强大的工具。此外，随机性是离散结构极端问题研究的一个主要主题，因为深入理解确定性对象模仿随机对应对象的方式通常会导致重大进展。这项研究有潜力通过开发用于大型网络计算问题的新算法、分析现有网络算法的新方法以及新的代码和通信协议来造福社会。该项目还为本科生和研究生提供培训机会。这项研究是在概率和极值组合学的广泛领域。概率组合学的研究主要集中在二项随机图的全局参数的高度集中和均匀随机图的边缘集分解为团或双团的问题上。在二项随机图的急剧集中的工作是由最近的研究人员和一个博士生的结果，建立了二项随机图的独立数的2点集中在一个广泛的概率参数范围。全面了解二项随机图独立数的集中程度是本部分研究计划的目标之一。本项目还包括进一步研究引人入胜的孤独奔跑者猜想以及关于超图和偏集的Ramsey数的一些问题。虽然我们在这两种情况下考虑的结构不一定是随机的，但我们希望结构和随机性的相互作用（即离散结构的伪随机性质）发挥主要作用。本研究的最终目标是寻找广泛适用于离散数学的新方法。该奖项反映了美国国家科学基金会的法定使命，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 research project is an investigation of discrete mathematical objects like networks and codes. Extremal combinatorics is focused on developing a better understanding of discrete mathematical objects that optimize interesting or desirable properties, while probabilistic combinatorics studies discrete mathematical objects that are generated by a series of random choices. These two research directions are intimately related as randomized algorithms are a remarkably powerful tool for the construction of interesting discrete mathematical objects. Furthermore, randomness is a major theme of the work on extremal problems for discrete structures as a deep understanding of the ways in which deterministic objects mimic their randomized counterparts often leads to major progress. This research has the potential to benefit society through the development of new algorithms for computational problems on large networks, new methods for analyzing existing network algorithms, and new codes and communication protocols. The project also provides training opportunities at both the undergraduate and graduate level.This research is in the broad areas of probabilistic and extremal combinatorics. The work in probabilistic combinatorics is focused on very sharp concentration of global parameters of the binomial random graph and problems regarding the decomposition of the edge set of the uniform random graph into cliques or bicliques. The work on sharp concentration in the binomial random graph is motivated by a recent result of the investigator and a doctoral student that establishes 2-point concentration of the independence number of the binomial random graph over a broad range of the probability parameter. The comprehensive understanding of the extent of concentration of the independence number of the binomial random graph is one of the goals of this part of the research program. This project also includes further study of the fascinating lonely runner conjecture and some problems on Ramsey numbers for hypergraphs and posets. While the structures that we consider in these two contexts are not necessarily random, we expect the interplay of structure and randomness (i.e. pseudorandom properties of discrete structures) to play a major role. The ultimate goal of this research is to find new methods that are broadly applicable in discrete mathematics.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.