Problems in Ramsey theory

拉姆齐理论中的问题

• 批准号: 2582036
• 金额: --
• 依托单位: University College London
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2025
• 资助国家: 英国
• 起止时间: 2025 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2582036
• 关键词: Problems; Ramsey; theory

In this proposal I will mention two future research directions that I hope to pursue together with my future Phd student. Both directions are in Ramsey theory. Ramsey's classical theorem (1930) asserts that whenever the edges of a complete graph G on N vertices are red-blue coloured, one can find a monochromatic (namely, all-red or all-blue) copy of a given graph H, provided that N is sufficiently large with respect to H. An interesting direction of research explores the structure of graphs G whose every red-blue colouring contains a monochromatic copy of H, and which are *minimal* with respect to this property. In particular, one could explore how small the minimum degree of such G could be. While a lot is known about this problem, there are many interesting open questions. Another direction with quite a different flavour explores the number of monochromatic copies of H in a red-blue coloured complete graph G. More specifically, one could ask for the maximum number of edge-disjoint monochromatic copies of H. Together with Gruslys, we have recently answered this question for the case where H is a triangle. Many variants of this question are open, yet some may be approachable with our techniques.Research Area: Logic and combinatorics

在此提案中，我将提到两个未来的研究指示，我希望与未来的博士生一起追求。这两个方向都在拉姆西理论中。拉姆齐的古典定理（1930）断言，每当n位顶点上完整的图形g的边缘是红蓝色的，就可以找到给定图H的单色（即全红或全蓝色）副本，前提*关于此属性的最小 *。特别是，人们可以探索这种G的最低程度可能是多少。虽然对这个问题有很多了解，但有许多有趣的开放问题。另一个带有不同风味的方向探索了红色彩色完整图G中H的单色副本的数量。更具体地说，可以要求最大数量的H. HH。与Gruslys一起，我们最近回答了这个问题，因为HES是h的情况。这个问题的许多变体都是开放的，但是我们的技术可能可以接近一些。研究领域：逻辑和组合。