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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=710081
• 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.

我提出的研究主要集中在两个数学领域：拉姆齐理论和极值图论。 拉姆齐理论是组合学的一个领域，从某种意义上说，它是由 20 世纪早期的逻辑和几何学工作发起的，它现在包括图论和数论的结果，基本上是一个典型的“拉姆齐-”。 “类型”结果表明，对于许多类型的结构（例如自然数或图形），任何大型结构都具有这样的属性：每当某些子结构被划分时，就可以在一个部分中找到“同质”模式。举一个例子，如果这自然数被分为两个集合，然后根据范德瓦尔登的定理，其中一个集合中存在任意长的算术级数。 拉姆齐理论现在已经相当发达，并在许多领域得到了应用，从图论到几何，拉姆齐类型的问题激发了（并促进了）许多数学领域的高级研究，包括数论、格论、集合论。 、拓扑学、理论计算机科学和概率，仅举几例，我打算专注于图论中的拉姆齐理论，以及与数字的一些有趣的联系，包括素数和加法组合。 在数学中，一个称为“图论”的领域不处理函数图（如抛物线），而是处理由“顶点”（点）和顶点之间的“边”（连接）组成的“网络”。可用于对算法、工作分配、交通、计算机网络（包括互联网）、流行病学甚至社交网络进行建模。 图论的一个专业称为“极值图论”。在极值图论中，人们可能会问，在保证出现选定的（较小的）图之前，某个图必须有多“稠密”。例如，如果100 个顶点有超过 2500 条边（大约是 4950 条边的一半），保证会出现三角形。另一个核心问题是确定哪些图最密集，同时仍不包含所选的小图。 “极值图”可用于提供复杂性、数论或几何中的关键示例。极值图论的许多结果都源于组合数论中的问题，例如，1938 年，Erdös 询问从 1 到 n 可以有多少个数字。被选择的成对乘积都不同？这是通过检查某些极值图（禁止简单的四周期图作为子图）来回答的。 另一方面，有限几何或数论中的某些例子被用来为拉姆齐型问题或极值图论问题提供答案，这两个领域的研究的一个特点是，人们不仅可以学习（和使用）。数学的各个领域都有许多出色的成果，但这两个领域的成果也对许多其他研究领域做出了贡献。