### 数学代写|离散数学作业代写discrete mathematics代考|There is a light that never goes out

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• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|离散数学作业代写discrete mathematics代考|What is Ramsey Theory

Consider positive integer solutions to $x+y=z$. Of course, $1+1=2$ and $2+3=5$ work if we allow all positive integers. So, let’s try to break this by splitting the positive integers into two parts. In Ramsey theory, we typically use colors to describe the partitions, so we will have, say, red integers and blue integers.

Must we still have a solution to $x+y=z$ if we now require the integers to be in the same partition, i.e., the same color? Let’s see if we can avoid the property of one part of the partition having a solution to $x+y=z$. First, 1 and 2 must be different colors (since $1+1=2$ ) and, consequently, 4 must be the same color as 1 (since $2+2=4$ ). Let’s say that 1 and 4 are red and 2 is blue. Since $1+4=5$, we see that 5 must also be blue, and, consequently, 3 must be red (since $2+3=5$ ). But now 1,3 , and 4 are all red, so the Ramsey property persists.

Ramsey properties also exist on graphs. For example, if we take $n \geq 3$ vertices and connect every pair of vertices with an edge, we clearly have a triangle with all edges in the same partition. Can we partition the edges in such a way so that we no longer have a triangle with all edges in the same partition? The answer is no, provided we have at least 6 vertices. To see this,

isolate one vertex, say $X$. Using the colors red and blue, since $X$ is connected to at least 5 other vertices, we see that one of the colors must occur on at least 3 of the edges. Let $X$ be connected to each of vertices $A, B$, and $C$ with a blue edge. If any edge between any two of $A, B$, and $C$ is blue, then we have a blue triangle. Otherwise, all edges among $A, B$, and $C$ are red and we again have a monochromatic (red) triangle. So, as long as we have enough vertices (here, we have shown that 6 vertices suffice), we cannot avoid a monochromatic triangle. It is easy to show that 6 vertices is also necessary by considering only 5 vertices $V_{0}, V_{1}, \ldots, V_{4}$ and letting the edge between $V_{i}$ and $V_{j}$ be red if $j \equiv i+1(\bmod 5)$, and blue otherwise.

The above examples are the easiest cases of two of the most well-known theorems in Ramsey theory: Schur’s Theorem $(x+y=z)$ and Ramsey’s Theorem (graphs). If these intrigue you, then you will find other compelling results in the coming chapters.

## 数学代写|离散数学作业代写discrete mathematics代考|Notations and Conventions

We will use the following notations and conventions. Other symbols that are used appear on page xiii.

Since Ramsey theory deals with colorings, formally, for $r \in \mathbb{Z}^{+}$, an $r$ coloring of the elements of a set $S$ is a mapping $\chi: S \rightarrow T$, where $|T|=r$ and, typically, $T={0,1, \ldots, r-1}$.

Unless otherwise stated, the intervals we use are integer intervals. Hence, we rely on the notation
$$[1, n]={1,2, \ldots, n} \text {. }$$
We may refer to this interval by $\mathbb{Z}_{n}$ if we are doing arithmetic modulo $n$. Similarly, we will assume all arithmetic progressions are of integers unless otherwise stated.

As hinted at by the definition of an $r$-coloring, we will be dealing with sets often. For any set $S$, we use $|S|$ to denote the cardinality of $S$ and we use $\rho(S)$ to denote the power set of $S$.

In our asymptotic analysis, we will use the notations $O(n), o(1)$, and $\ll$. We remind the reader of these next.

Definition $1.1$ (Big-O, Little-o, and $\ll)$. We say that $f(n)=O(n)$ if there exists a constant $c>0$ such that $|f(n)| \leq c n$ for all sufficiently large $n$. We say that $f(n)=o(1)$ if $\lim {n \rightarrow \infty} f(n)=0$. We say $f(n) \leqslant g(n)$ if $\lim {n \rightarrow \infty} \frac{f(n)}{g(n)}=0$.
We will also be using logarithms often. For our purposes, log is the base 2 logarithm and $\ln$ is the natural logarithm.

## 数学代写|离散数学作业代写discrete mathematics代考|Combinatorics

When doing asymptotic analysis, we will rely on two results. The first is Stirling’s formula, formulated as
$$n ! \approx \sqrt{2 \pi n}\left(\frac{n}{e}\right)^{n}$$
The second is
$$\ln (1+x) \approx x$$
for $x$ small. Some consequences of these appear in the Exercises section of this chapter, and those consequences will be used without reference.

Many results in Ramsey theory have their origins with the pigeonhole principle. This principle is obvious in statement, but not necessarily so in application. For reference, here is the pigeonhole principle.

Theorem $1.2$ (Pigeonhole Principle). Let $k, r \in \mathbb{Z}^{+} .$If $n \geq k r+1$ elements are partitioned into $r$ parts, then one of those parts must contain at least $k+1$ elements.

The last combinatorial concept we remind the reader of is the Principle of Inclusion-Exclusion. It is a counting principle but can also be stated in terms of probabilities (see Equation 6.1).

Theorem 1.3 (Principle of Inclusion-Exclusion). Let $S_{1}, S_{2}, \ldots, S_{m}$ be finite sets. Then
\begin{aligned} \left|\bigcup_{i=1}^{m} S_{i}\right| &=\sum_{i=1}^{m}\left|S_{i}\right|-\sum_{1 \leq i<j \leq m}\left|S_{i} \cap S_{j}\right|+\sum_{1 \leq i<j<k \leq m}\left|S_{i} \cap S_{j} \cap S_{k}\right|-\cdots+(-1)^{m+1}\left|\bigcap_{i=1}^{m} S_{i}\right| \ &=\sum_{i=1}^{m} \sum_{\substack{I \subseteq[1, m] \ |X|=i}}(-1)^{i+1}\left|\bigcap_{j \in I} S_{j}\right| \end{aligned}In Ramsey theory it is usually not feasible to calculate all terms in the inclusion-exclusion formula. Hence, we will use the following Bonferroni inequalities.

## 数学代写|离散数学作业代写discrete mathematics代考|What is Ramsey Theory

Ramsey 属性也存在于图上。例如，如果我们采取n≥3顶点并将每对顶点与一条边连接起来，我们显然有一个三角形，所有边都在同一个分区中。我们能否以这样的方式对边进行分区，以使我们不再有一个所有边都在同一个分区中的三角形？答案是否定的，只要我们至少有 6 个顶点。看到这个，

[1,n]=1,2,…,n.

## 数学代写|离散数学作业代写discrete mathematics代考|Combinatorics

n!≈2圆周率n(n和)n

ln⁡(1+X)≈X

|⋃一世=1米小号一世|=∑一世=1米|小号一世|−∑1≤一世<j≤米|小号一世∩小号j|+∑1≤一世<j<ķ≤米|小号一世∩小号j∩小号ķ|−⋯+(−1)米+1|⋂一世=1米小号一世| =∑一世=1米∑一世⊆[1,米] |X|=一世(−1)一世+1|⋂j∈一世小号j|在拉姆齐理论中，计算包含排除公式中的所有项通常是不可行的。因此，我们将使用以下 Bonferroni 不等式。

## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。