Hypergraphs, Ramsey Theory and Extremal Combinatorics

超图、拉姆齐理论和极值组合

• 批准号: 1301698
• 负责人: Vojtech Rodl
• 金额: $ 28.51万
• 依托单位: Emory University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-06-01 至 2018-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1301698&HistoricalAwards=false
• 关键词: Hypergraphs; Ramsey; Theory; Extremal; Combinatorics

The proposed research concentrates mainly on problems in Ramsey theory and on extremal graph and hypergraph theory. Both of these areas belong to the mainstream of contemporary discrete mathematics and are often motivated by applications in other areas, such as theoretical computer science. In tackling these problems, a variety of combinatorial methods will be used, such as probabilistic methods and applications of regularity techniques. The PI also plans to develop new techniques leading to the construction of graphs with certain special properties. The Ramsey type problems suggested in the proposal include the investigation of the asymptotic behavior of hypergraph Ramsey numbers, of induced Ramsey numbers, of size-Ramsey numbers, as well of the Erdos-Rogers function. The PI also intends to study some problems in structural Ramsey theory. The extremal problems that the PI intends to investigate include Ramsey-Turán problems and the study of Hamilton cycles and matchings in hypergraphs, as well as extremal problems regarding random subsets of integers.This research will serve as a basis for the mentoring activities of the PI. Concepts and questions of combinatorial character appear naturally in other branches of mathematics and, more broadly speaking, in computer science and in the physical, biological, and social sciences, making combinatorial mathematics of interest not only within mathematics itself, but within the sciences as a whole.

拟议的研究主要集中在拉姆齐理论以及极值图和超图理论中的问题，这两个领域都属于当代离散数学的主流，并且通常受到其他领域（例如理论计算机科学）的应用的推动。 ，将使用各种组合方法，例如概率方法和规律性技术的应用，PI 还计划开发新技术，以构建具有某些特殊属性的图，其中包括调查。的超图拉姆齐数、诱导拉姆齐数、大小拉姆齐数以及 Erdos-Rogers 函数的渐近行为 PI 还打算研究结构 Ramsey 理论中的一些问题 PI 打算研究的极值问题包括。 Ramsey-Turán 问题以及超图中汉密尔顿循环和匹配的研究，以及有关整数随机子集的极值问题。这项研究将作为指导活动的基础组合特征的概念和问题自然地出现在数学的其他分支中，更广泛地说，出现在计算机科学以及物理、生物和社会科学中，使得组合数学不仅在数学本身中，而且在科学中也引起人们的兴趣。作为一个整体。