Extremal graph theory and Ramsey theory

极值图论和拉姆齐理论

• 批准号: RGPIN-2016-05959
• 负责人: Gunderson, David
• 金额: $ 1.09万
• 依托单位: University of Manitoba
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=616015
• 关键词: Extremal; graph; theory; Ramsey

My proposed research is primarily in two areas of mathematics, Ramsey theory and extremal graph theory. Ramsey theory is an area of combinatorics that was, in a sense, initiated by work in logic and, independently, geometry early in the 20th century. It now includes results from graph theory and number theory. Basically, a typical "Ramsey-type" result says that for many classes of structures (e.g., the natural numbers, or graphs) any large structure has the property that whenever certain substructures are partitioned, "homogeneous" patterns can be found in one part. To give one example, if the natural numbers are partitioned into two sets, then by a theorem due to van der Waerden, there exist arbitrarily long arithmetic progressions in one of the sets. Ramsey theory is now rather well-developed, and has since found application in many fields, from graph theory to geometry. Ramsey type problems have inspired (and contributed to) advanced research in many mathematical fields, including number theory, lattice theory, set theory, topology, theoretical computer science and probability, to name but a few. I intend to concentrate on Ramsey theory in graph theory, and on some interesting connections with numbers, including the primes and additive combinatorics. In mathematics, an area called "graph theory" does not deal with graphs of functions (like parabolas), but instead deals with "networks" consisting of "vertices" (points) and "edges" (connections) between vertices. Modern graphs can be used to model algorithms, job allocations, transportation, computer networks (including the internet), epidemiology, or even social networks. One specialization in graph theory is called "extremal graph theory''. In extremal graph theory, one might ask how "dense" a certain graph must be before a chosen (smaller) graph is guaranteed to appear. For example, if a graph on 100 vertices has more than 2500 edges (about half of the 4950 possible) a triangle is guaranteed to appear. Another central question is to determine which graphs are the densest while still not containing the chosen small graph. Such "extremal graphs" can be used to provide critical examples in complexity, number theory, or geometry. Many results in extremal graph theory arose from questions in combinatorial number theory. For example, in 1938, Erdös asked how many numbers from 1 to n can be chosen whose pairwise products are all different? This was answered by examining certain extremal graphs (that forbid a simple four-cycle graph as a subgraph). On the other hand, certain examples from finite geometries or number theory are used to provide answers to Ramsey-type problems or extremal graph theory problems. One feature of studying in these two areas is that not only does one get to learn (and use) many fantastic results from various areas of mathematics, but results in these two areas also contribute back to so many other fields of research.

我的支撑研究主要在数学，拉姆西理论和极端图中的两个领域。 拉姆西理论是由逻辑上的工作和独立的几何形状开始的组合学区域。许多级别的结构（自然数或图）都可以分配任何大型结构，“同质”模式可以在一个部分中找到一个例子，如果自然数被分配为两组，那么由于范德登（Van Derden）引起的定理，其中一个集合中存在任意长的算术进程。 拉姆西（Ramsey）现在的评估者已经发达了，此后在许多数学领域中发现了（并促成）许多数学领域，倾斜数量理论，晶格理论，集合理论，拓扑，理论计算机，促进了许多领域的应用。科学和概率，姓名，但专注于拉姆西（Ramsey），在图理论中与数字相关联，使素数和固定组合学不一致。 在数学中，一个称为“ dex不涉及函数图的区域（例如抛物线），而是处理“顶点”（点）和“边缘”（连接）的“网络” ting。模型算法，作业分配，运输，计算机网络（包括互联网）甚至社交网络。 理论上的一种专业化被称为“极端图理论”。三角形的pH值是pHS，而不是选择的“极端图”。例如，在1938年，埃德斯（Erdös）询问了从1到n的数字，可以通过检查某些极端图（禁止简单的四个周期图作为ASUBGRAPH）来回答这一点。 另一方面，有限的几何形状或数字的某些例子。回到许多其他研究领域。