拉姆齐理论专题
Topics in Ramsey Theory
摘要：
拉姆齐理论（Ramsey theory）是一门研究数学对象在被分割时表现出有序性的理论。关于整数的拉姆齐理论关注$[1,n]$到$r$子集的分区本身，并询问这些$r$子集中的一个（或多个）是否包含$\mathcal{F}$的$k$ -项成员的问题，其中$[1,n]=\{1,2,3,\ldots,n\}$和$\mathcal{F}$是$\mathbb{Z}^+$的子集的某个族。当$\mathcal{F}$固定为算术级数的集合时，对应的Ramsey型数称为van der Waerden数。我在项目开始时选择$\mathcal{F}$作为范围$m$的半进展集。作用域$m\in \mathbb{Z}^+$的半级数是一组整数$\{x_1,x_2,\ldots,x_k\}$，使得对于某些$d\in\mathbb{Z}^+$，对于所有$i\in\{2,3,\ldots,k\}$，为$x_{i}-x_{i-1}\in\{d,2d,\ldots,md\}$。对应于半级数的拉姆齐型函数的精确值是未知的。我们使用$SP_m(k)$来表示这些数字，作为$k$的Ramsey型函数，用于固定范围的$m$。在这个项目中，我使用概率方法来获得任何固定$m$的指数下限。第一章首先简要介绍了拉姆齐理论，然后解释了所考虑的问题。在第二章中，我们给出了半级数的一些结果。在第三章中，我们将讨论$Q_1(k)$上的下界。当选择$\mathcal{F}$作为直径$n$的拟级数时，得到的相应Ramsey型数记为$Q_n(k)$。最后一章阐述了先进的概率技术，特别是浓度不等式和如何应用它们。
英文摘要：
Ramsey theory is the study of conditions under which mathematical objects show order when partitioned. Ramsey theory on the integers concerns itself with partitions of $[1,n]$ into $r$ subsets and asks the question whether one (or more) of these $r$ subsets contains a $k$-term member of $\mathcal{F}$, where $[1,n]=\{1,2,3,\ldots,n\}$ and $\mathcal{F}$ is a certain family of subsets of $\mathbb{Z}^+$. When $\mathcal{F}$ is fixed to be the set of arithmetic progressions, the corresponding Ramsey-type numbers are called the van der Waerden numbers. I started the project choosing $\mathcal{F}$ to be the set of semi-progressions of scope $m$. A semi-progression of scope $m\in \mathbb{Z}^+$ is a set of integers $\{x_1,x_2,\ldots,x_k\}$ such that for some $d\in\mathbb{Z}^+$, $x_{i}-x_{i-1}\in\{d,2d,\ldots,md\}$ for all $i\in\{2,3,\ldots,k\}$. The exact values of Ramsey-type functions corresponding to semi-progressions are not known. We use $SP_m(k)$ to denote these numbers as a Ramsey-type function of $k$ for a fixed scope $m$. During this project, I used the probabilistic method to get an exponential lower bound for any fixed $m$. The first chapter starts with a brief introduction to Ramsey theory and then explains the problem considered. In the second chapter, I give the results obtained on semi-progressions. In the third chapter, I will discuss the lower bound obtained on $Q_1(k)$. When $\mathcal{F}$ is chosen to be quasi-progressions of diameter $n$, the corresponding Ramsey-type numbers obtained are denoted as $Q_n(k)$. The last chapter gives an exposition of advanced probabilistic techniques, in particular concentration inequalities and how to apply them.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
