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

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## 数学代写|离散数学作业代写discrete mathematics代考|Van der Waerden’s Theorem

The most well-known result in Ramsey theory on the integers is van der Waerden’s Theorem [205] concerning arithmetic progressions.

Theorem 2.1 (van der Waerden’s Theorem). Let $k, r \in \mathbb{Z}^{+}$. There exists a minimal positive integer $w(k ; r)$ such that for any $n \geq w(k ; r)$ the following holds: for any $r$-coloring of $[1, n]$ there exist $a, d \in \mathbb{Z}^{+}$such that
$$a, a+d, a+2 d, \ldots, a+(k-1) d$$
is a monochromatic $k$-term arithmetic progression.
This theorem, along with Ramsey’s Theorem (Theorem 3.6), are the most famous of Ramsey-type theorems. Van der Waerden [205] had credited Baudet with the conjecture that Theorem $2.1$ is true (the translated title of van der

Waerden’s article is “Proof of a Baudet Conjecture”). However, extensive historical research done by Soifer is presented in his fascinating book [191] and provides compelling evidence that Baudet and Schur (who we will meet in Section 2.2.1) independently made the same conjecture.

The proof we present here is elementary. Although these proofs are easy to find (see, e.g., $[83,84,119,129]$ ) this book would be incomplete without presenting one such proof. Other “non-elementary” proofs have been done as we will see in subsequent chapters of this book.

Proof of Theorem $2.1$. We will prove that $w(k ; r)$ exists by inducting on $k$, with $w(2 ; r)=r+1$ being the base case, which holds by the pigeonhole principle. Hence, we may assume that $w(k-1 ; r)$ exists for all $r$ to prove that $w(k ; r)$ exists for all $r$.

We will say that a set of monochromatic $(k-1)$-term arithmetic progressions $a_{i}+\ell d_{i}, 0 \leq \ell \leq k-2$, are end-focused if they are each of a different color and $a_{i}+(k-1) d_{i}=a_{j}+(k-1) d_{j}$ for all $i$ and $j$; that is, when each of the arithmetic progressions is extended one more term, those last terms all coincide.

We will prove the induction step by showing that for any $s \in \mathbb{Z}^{+}$with $s \leq r$ there exists an integer $n=n(k, s ; r)$ such that every $r$-coloring of $[1, n]$ either contains a monochromatic $k$-term arithmetic progression or contains $s$ end-focused $(k-1)$-term arithmetic progressions. To see that the existence of $n(k, s ; r)$ for all $s \leq r$ proves the existence of $w(k ; r)$, note that for $n(k, r ; r)$, one of the end-focused arithmetic progressions of length $k-1$ extends to a monochromatic $k$-term arithmetic progression (since all end-focused arithmetic progressions have different colors).

To prove the existence of $n(k, s ; r)$ we induct on $s$, noting that $n(k, 1 ; r)=$ $w(k-1 ; r)$ works, which exists by our induction on $k$ assumption. We now assume that $n=n(k, s-1 ; r)$ exists, in addition to the existence of $w(k-1, r)$ for all $r$. Hence, we may consider $m=4 n w\left(k-1 ; r^{2 n}\right)$.

## 数学代写|离散数学作业代写discrete mathematics代考|Hilbert’s Cube Lemma

Arguably the first Ramsey-type result is due to Hilbert [108] as it adheres to the partitioning ethos of Ramsey theory. It should be noted that Hilbert’s goal in using this lemma was a result about the irreducibility of polynomials; see $[209]$ for more information. To describe the cube referenced in the title of this subsection, we make the following definition.

Definition $2.9$ (Finite sums). The set of finite sums of integers $x_{1}, x_{2}, \ldots, x_{n}$ is denoted and defined as
$$F S\left(x_{1}, x_{2}, \ldots, x_{n}\right)=\left{\sum_{i \in I} x_{i}: \emptyset \neq I \subseteq[1, n]\right} .$$
Note that in the above definition, the integers $x_{i}$ are not necessarily distinct.
We may now define the aforementioned cube.
Definition $2.10$ (d-cube). The $d$-cube of integers $c, x_{1}, x_{2}, \ldots, x_{d}$ is the set of integers $c+F S\left(x_{1}, x_{2}, \ldots, x_{n}\right)$.

Before getting to Hilbert’s result, a little bit of intuition into why this is named the cube lemma is warranted. Consider the unit cube in $\mathbb{R}^{3}$ with vertices $\left{\left(\epsilon_{1}, \epsilon_{2}, \epsilon_{3}\right): \epsilon_{i} \in{0,1}\right}$. Let $c=(0,0,0), x_{1}=(1,0,0), x_{2}=(0,1,0)$, and $x_{3}=(0,0,1)$. Then the other vertices are $x_{1}+x_{2}, x_{1}+x_{3}$, and $x_{1}+x_{2}+x_{3}$. In other words, the vertices of the unit cube are $c+F S\left(x_{1}, x_{2}, x_{3}\right)$. This is easily abstracted to higher dimensions and different side lengths.
We now state Hilbert’s result.
Lemma 2.11 (Hilbert’s Cube Lemma). Let $r, d \in \mathbb{Z}^{+} .$Any $r$-coloring of $\mathbb{Z}^{+}$ admits a monochromatic $d$-cube.

This lemma follows from van der Waerden’s Theorem since a $(d+1)$-term arithmetic progression $a, a+\ell, \ldots, a+d \ell$ is a $d$-cube of $a, \ell, \ell, \ldots, \ell$; however, we will give an independent proof.

Proof. We induct on $d$, with $d=1$ being trivial since we only need 2 integers of the same color. Assume the result for $d-1$. By the Compactness Principle there exists an integer $h=h(d-1 ; r)$ such that any $r$-coloring of $[1, h]$ admits a monochromatic $(d-1)$-cube. Consider any $r$-coloring of $\left[1,\left(r^{h}+1\right) h\right]$. There are $r^{h}$ possible colorings of any interval of length $h$ and we have $r^{h}+1$ disjoint intervals of length $h$, namely, $[(i-1) h+1, i h]$ for $1 \leq i \leq r^{h}+1$. Hence, two such intervals, say $[(a-1) h+1, a h]$ and $[(b-1) h+1, b h]$, with $a<b$, are colored identically.

We claim that any $r$-coloring of a translated interval of length $h$ also admits a monochromatic $(d-1)$-cube. Via the obvious bijection between $[1, h]$ and $[n+1, n+h]$, the color of any $c+\sum_{i \in I} x_{i}$ in $[1, h]$ is the same as the color of $(c+n)+\sum_{i \in I} x_{i}$ in $[n+1, n+h]$. Hence, if $c+F S\left(x_{1}, x_{2}, \ldots, x_{d-1}\right)$ is a monochromatic $(d-1)$-cube in $[1, h]$ then $(c+n)+F S\left(x_{1}, x_{2}, \ldots, x_{d-1}\right)$ is a monochromatic $(d-1)$-cube in $[n+1, n+h]$.

With this translation invariance, we apply the inductive assumption to see that the coloring of $[(a-1) h+1, a h]$ admits a monochromatic $(d-1)$-cube of $c, x_{1}, x_{2}, \ldots, x_{d-1}$. Now consider the $d$-cube of $c, x_{1}, x_{2}, \ldots, x_{d-1},(b-a) h$. Because $[(a-1) h+1, a h]$ and $[(b-1) h+1, b h]$ are identically colored, we see that $c+F S\left(x_{1}, x_{2}, \ldots, x_{d-1},(b-a) h\right)$ is a monochromatic $d$-cube. This completes the induction.

## 数学代写|离散数学作业代写discrete mathematics代考|Deuber’s Theorem

A useful result with somewhat the same flavor as Hilbert’s Cube Lemma was given by Deuber [56] in 1973 via repeated application of van der Waerden’s Theorem. In order to state the result, we require a definition.

Definition $2.12((m, p, c)$-set $)$. Let $m, p, c \in \mathbb{Z}^{+} . \mathrm{A}$ set $M \subseteq \mathbb{Z}^{+}$is called an $(m, p, c)$-set if there exist generators $x_{0}, x_{1}, x_{2}, \ldots, x_{m} \in \mathbb{Z}^{+}$such that
$$M=\bigcup_{i=0}^{m}\left{c x_{i}+\sum_{j=i+1}^{m} \lambda_{j} x_{j}: \lambda_{j} \in \mathbb{Z} \cap[-p, p] \text { for } 1 \leq j \leq m\right}$$
where we take the empty sum to equal 0 .
Note that we have the integers $\lambda_{j}$ taking on negative values but that the set $M$ must be a set of positive integers.

We may now state Deuber’s result. Notice how this result shows the unbreakable property of $(m, p, c)$-sets.

Theorem 2.13 (Deuber’s Theorem). Let $m, p, c, r \in \mathbb{Z}^{+}$be fixed. Then thene exist $M, P, C \in \mathbb{Z}^{+}$so that every $r$-coloring of an arbitrary $(M, P, C)$-set admits a monochromatic $(m, p, c)$-set.

We will not need the full strength of Deuber’s Theorem; rather, the following weaker version (with a more digestible proof) will be useful for us.
Theorem $2.14$ (Weak Deuber’s Theorem). Let $m, p, c, r \in \mathbb{Z}^{+}$be fixed. Every $r$-coloring of $\mathbb{Z}^{+}$admits a monochromatic $(m, p, c)$-set.

## 数学代写|离散数学作业代写discrete mathematics代考|Van der Waerden’s Theorem

Ramsey 整数理论中最著名的结果是关于算术级数的范德瓦尔登定理 [205]。

Waerden 的文章是“鲍德猜想的证明”）。然而，Soifer 所做的广泛历史研究在他引人入胜的著作 [191] 中进行了介绍，并提供了令人信服的证据，证明 Baudet 和 Schur（我们将在第 2.2.1 节中遇到他们）独立地做出了同样的猜想。

## 数学代写|离散数学作业代写discrete mathematics代考|Hilbert’s Cube Lemma

F S\left(x_{1}, x_{2}, \ldots, x_{n}\right)=\left{\sum_{i \in I} x_{i}: \emptyset \neq I \subseteq[1 , n]\right} 。F S\left(x_{1}, x_{2}, \ldots, x_{n}\right)=\left{\sum_{i \in I} x_{i}: \emptyset \neq I \subseteq[1 , n]\right} 。

## 数学代写|离散数学作业代写discrete mathematics代考|Deuber’s Theorem

Deuber [56] 在 1973 年通过反复应用范德瓦尔登定理给出了一个与希尔伯特立方引理有点相似的有用结果。为了说明结果，我们需要一个定义。

M=\bigcup_{i=0}^{m}\left{c x_{i}+\sum_{j=i+1}^{m} \lambda_{j} x_{j}: \lambda_{j} \in \mathbb{Z} \cap[-p, p] \text { for } 1 \leq j \leq m\right}M=\bigcup_{i=0}^{m}\left{c x_{i}+\sum_{j=i+1}^{m} \lambda_{j} x_{j}: \lambda_{j} \in \mathbb{Z} \cap[-p, p] \text { for } 1 \leq j \leq m\right}

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