Probabilistic and Extremal Combinatorics

本研究项目是对离散数学对象（如网络和代码）的研究。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生成的离散数学对象。这两个研究方向是密切相关的随机算法是一个非常强大的工具，为建设有趣的离散数学对象。此外，随机性是离散结构极值问题工作的一个主要主题，因为对确定性对象模仿随机对象的方式的深入理解往往会导致重大进展。这项研究有可能通过开发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。该项目还提供了本科生和研究生的培训机会。这项研究是在概率和极值组合学的广泛领域。在概率组合学的工作集中在非常尖锐的浓度的全球参数的二项式随机图和问题的分解的边缘集的均匀随机图成团或bicliques。在二项式随机图中的浓度急剧的工作是由最近的结果的调查员和一个博士生，建立2点浓度的独立数的二项式随机图在广泛的概率参数。全面了解二项随机图的独立数的集中程度是这部分研究计划的目标之一。本项目还包括对有趣的孤独跑步者猜想的进一步研究以及超图和偏序集的Ramsey数的一些问题。虽然我们在这两个上下文中考虑的结构不一定是随机的，但我们希望结构和随机性（即离散结构的伪随机特性）的相互作用发挥主要作用。该研究的最终目标是找到广泛适用于离散数学的新方法。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 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.