数学代写|离散数学作业代写discrete mathematics代考|Compactness Principle

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数学代写|离散数学作业代写discrete mathematics代考|Compactness Principle

There is an interplay between finite and infinite Ramsey theory. While much of this book focuses on finite Ramsey theory, we can use the infinite to prove the finite. This is accomplished by the Compactness Principle, which we state below.

Compactness Principle. Let $\mathcal{F}$ be a family of finite subsets of $\mathbb{Z}^{+}$. Let $k, r \in \mathbb{Z}^{+}$. Assume that every $r$-coloring of the $k$-element subsets of $\mathbb{Z}^{+}$admits $F \in \mathcal{F}$ with the property that all $k$-element subsets of $F$ have the same color. Then there exists $N \in \mathbb{Z}^{+}$such that for all $n \geq N$, any $r$-coloring of the $k$ element subsets of $[1, n]$ admits $G \in \mathcal{F}$ with $G \subseteq[1, n]$ such that the collection of $k$-element subsets of $G$ is monochromatic.

This result can be used at times to bypass technical details in proofs. Since many Ramsey theory results are about having “large enough” systems, when dealing with the set of integers we often encounter statements of the form “for all $n \geq N$, property $P$ holds.” If we can show that property $P$ holds over the positive integers, then the Compactness Principle gives us the “for all $n \geq N “$ part of the statement.

The proof of the Compactness Principle is essentially Cantor’s argument for proving that the set of real numbers is uncountable; the reader is referred to $[129]$ for a proof.

A word of warning is warranted here. The Compactness Principle does not work in reverse; that is, we cannot prove the finite version and conclude that it holds for the infinite. As we will see, results on infinite sets can run counter to their finite counterparts. Just keep in mind that arbitrarily large does not mean infinite.

数学代写|离散数学作业代写discrete mathematics代考|Set Theoretic Considerations

We will be focusing our attention on countable objects; however, Ramsey theory is also studied over uncountable sets. We can ask, for example, do similar results hold if we color $\mathbb{R}^{+}$instead of $\mathbb{Z}^{+} ?$ When doing so, the Axiom of Choice (or one of its equivalents) may come into play. We will largely (but not always) stay away from this uncountable territory; see [117] for a recent treatment of Ramsey theory in the uncountable setting.

We will note here that, like the Banach-Tarski paradox, we get some strange results in the uncountable setting. Consider the set of infinite subsets of $\mathbb{Z}^{+}$, denoted by $\mathcal{P}$. Assume that each $P \in \mathcal{P}$ is assigned one of two colors. In order for $\mathcal{P}$ to have the Ramsey property, we would require that

under every possible 2 -coloring $\chi$ of the elements of $\mathcal{P}$, there exists an infinite set $S \in \mathcal{P}$ such that all infinite subsets of $S$ have the same color under $\chi$.
As noted by Galvin and Prikry [79], both of the following hold:
(i) assuming the Axiom of Choice is true (so that there exist subsets of $\mathbb{R}$ that are non-measurable), Erdôs and Rado [68] have provided a 2 coloring of the elements of $\mathcal{P}$ such that no infinite set has all infinite subsets the same color;
(ii) assuming the Axiom of Choice is false and that all subsets of $\mathbb{R}$ are measurable, Mathias [142] has shown that under any 2-coloring of the elements of $\mathcal{P}$ there exists an infinite set that has all infinite subsets the same color.

By restricting ourselves to countable sets, we will focus on the Ramseytheoretic content of the material and not on the set-theoretic aspects, which, as we see above, can lead to “paradoxical” results.

There are a few instances in this book where we do appeal to the Axiom of Choice in the equivalent form of Zorn’s Lemma, which we now state.

Zorn’s Lemma. If every chain in a partially ordered set $S$ has an upper bound (respectively, lower bound) in $S$, then $S$ contains a maximal (respectively, minimal) element.

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

$1.1$ Show that
$$\left(1+\frac{x}{n}\right)^{n} \approx e^{x}$$
for large $n$.
$1.2$ Let $k, n \in \mathbb{Z}^{+}$be large, with $n \gg k$. Show that
$$\left(\begin{array}{l} n \ k \end{array}\right) \approx \frac{1}{\sqrt{2 \pi k}} \cdot\left(\frac{n e}{k}\right)^{k}$$
$1.3$ Consider the $k$-element sets of $[1, n]$. If we color each integer in $[1, n]$ randomly with one of $r$ colors, what is the probability that a particular $k$-element set is monochromatic? What is the probability that at least one of the $k$-element sets is monochromatic if $n>r(k-1)$ ?
$1.4$ Show that for any $n+1$ integers chosen from ${1,2, \ldots, 2 n}$, two of the chosen integers have the property that one divides the other.

$1.5$ Let $n \in \mathbb{Z}^{+}$and consider a set $S$ of $n$ integers, none of which are divisible by $n$. Show that there exists a subset $\emptyset \neq T \subseteq S$ such that $n$ divides $\sum_{t \in T} t$.
$1.6$ Let $n \in \mathbb{Z}^{+}$. Show that there are two powers of two that differ by a multiple of $n$.
$1.7$ This is a gem due to Erdôs and Szekeres [70]. Let $n, m \in \mathbb{Z}^{+}$. Prove that every sequence of $n m+1$ distinct numbers contains either an increasing subsequence of length $n+1$ or a decreasing subsequence of length $m+1$.
Hint:
For each number in the sequence let $\ell_{i}$ be the length of the longest increasing subsequence starting at the $i^{\text {th }}$ term.
$1.8$ How many arithmetic progressions of length 3 are contained in $[1, n]$ ? How many of length $k$ ?
$1.9$ Let $S$ be a set with $|S|=n$. Let $F$ be the set of bijections from $S$ to $S$ such that for $f \in F$ we have $f(s) \neq s$ for all $s \in S$. Use the Principle of Inclusion-Exclusion to show that $|F|$ is the integer nearest $\frac{n !}{e}$.
$1.10$ For $t \in \mathbb{Z}{n}$, let $$\chi(t)=e^{\frac{2 \pi i t}{n}},$$ where $i=\sqrt{-1}$. Prove directly that, for $t \neq 0$ we have $$\sum{k=0}^{n-1} \chi(k t)=0$$
$1.11$ Consider the function $f$ defined on $\mathbb{Z}{5}$ by $f(x)=x^{2}$. Find $\widehat{f}(t)$, the discrete Fourier transform of $f$, and verify (i) and (iii) of Theorem 1.9. $1.12$ Consider all infinite binary strings. For two strings $s{1} s_{2} s_{3} \ldots$ and $t_{1} t_{2} t_{3} \ldots$, let $n$ be the minimal positive integer for which $s_{n} \neq t_{n}$. Show that $d(s, t)=2^{-n}$ defines a metric on the space of infinite binary strings.
1.13 Prove that Corollary $1.7$ follows from Lemma 1.6.
$1.14$ Let $n \in \mathbb{Z}^{+}$. Let $S$ be a strict subspace of $\mathbb{Q}^{n}$. Define $S^{\perp}$ to be the orthogonal complement of $S$. Describe $S^{\perp}$ and prove that every element of $\mathbb{Q}^{n}$ can be written as $s+s^{\perp}$ for some $s \in S$ and $s^{\perp} \in S^{\perp}$.
$1.15$ Let $p$ be prime and let $n \in \mathbb{Z}^{+}$. How many $x \in{1,2, \ldots, p-1}$ satisfy $x^{n} \equiv 1(\bmod p) ?$

数学代写|离散数学作业代写discrete mathematics代考|Set Theoretic Considerations

（i）假设选择公理是正确的（因此存在R是不可测量的），Erdôs 和 Rado [68] 提供了 2 种颜色的元素磷使得没有无限集的所有无限子集都具有相同的颜色；
(ii) 假设选择公理是错误的并且所有子集R是可测量的，Mathias [142] 表明，在元素的任何 2 着色下磷存在一个无限集，它的所有无限子集都具有相同的颜色。

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

1.1显示
(1+Xn)n≈和X

1.2让ķ,n∈从+大，有n≫ķ. 显示
(n ķ)≈12圆周率ķ⋅(n和ķ)ķ
1.3考虑ķ-元素集[1,n]. 如果我们为每个整数着色[1,n]随机与其中之一r颜色，一个特定的概率是多少ķ-元素集是单色的？至少有一个的概率是多少ķ- 元素集是单色的，如果n>r(ķ−1) ?
1.4显示任何n+1从中选择的整数1,2,…,2n，两个选定的整数具有一个除另一个的性质。

1.5让n∈从+并考虑一个集合小号的n整数，其中任何一个都不能被n. 证明存在一个子集∅≠吨⊆小号这样n划分∑吨∈吨吨.
1.6让n∈从+. 证明有两个相差 的倍数的 2 的幂n.
1.7这是 Erdôs 和 Szekeres [70] 的瑰宝。让n,米∈从+. 证明每一个序列n米+1不同的数字包含一个增加的长度子序列n+1或长度递减的子序列米+1.

1.8长度为 3 的等差数列包含多少个[1,n]? 多少长ķ ?
1.9让小号与|小号|=n. 让F是来自的双射集小号到小号这样对于F∈F我们有F(s)≠s对全部s∈小号. 使用包含-排除原理来证明|F|是最接近的整数n!和.
1.10为了吨∈从n， 让χ(吨)=和2圆周率一世吨n,在哪里一世=−1. 直接证明，对于吨≠0我们有∑ķ=0n−1χ(ķ吨)=0
1.11考虑函数F定义于从5经过F(X)=X2. 寻找F^(吨), 的离散傅里叶变换F，并验证定理 1.9 的 (i) 和 (iii)。1.12考虑所有无限的二进制字符串。对于两个字符串s1s2s3…和吨1吨2吨3…， 让n是最小的正整数sn≠吨n. 显示d(s,吨)=2−n定义无限二进制字符串空间的度量。
1.13 证明推论1.7遵循引理 1.6。
1.14让n∈从+. 让小号是一个严格的子空间问n. 定义小号⊥是的正交补小号. 描述小号⊥并证明每个元素问n可以写成s+s⊥对于一些s∈小号和s⊥∈小号⊥.
1.15让p成为素数并让n∈从+. 多少X∈1,2,…,p−1满足Xn≡1(反对p)?

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