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

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

Thanks to Rado, we have a complete understanding of which systems of linear equations are regular. An obvious next step is the investigation of nonlinear equations.

Perhaps Pythagorean triples, i.e., solutions to $x^{2}+y^{2}=z^{2}$, pop to mind first. While this is a homogeneous equation, let’s not make the situation harder than we must. We’ll start with $x+y=z^{2}$. The following result is due to Green and Lindqvist $[88]$; we follow Pach $[156]$ for the proof of 2-regularity.
Theorem 2.28. The equation $x+y=z^{2}$ is 2-regular, but not 3-regular.
Proof. Of course $(x, y, z)=(2,2,2)$ is always a monochromatic solution under any coloring, so we do not allow this as a valid monochromatic solution.

We start by exhibiting a 3 -coloring of $\mathbb{Z}^{+}$with no monochromatic solution. Define the intervals
$$I_{j}=\left{i \in \mathbb{Z}^{+}: 2^{j} \leq i \leq 2^{j+1}-1\right} \quad \text { for } \quad j=0,1,2, \ldots$$
Let the colors be 0,1 , and 2 . For $j=0,1$, and 2 , color all elements of $I_{j}$ by color $j$. For $j=3,4, \ldots$, in order, color all elements of $I_{j}$ with a color missing from
$$I_{\left\lfloor\frac{j}{2}\right\rfloor} \cup I_{\left\lfloor\frac{j}{2}\right\rfloor+1} .$$

Assume, for a contradiction, that $a+b=c^{2}$ with $a \leq b$ is a monochromatic solution under this coloring. For some $j$ we have $b \in I_{j}$. Since $a \leq b$ we have $2^{j}<a+b<2^{j+2}$ so that $2^{\frac{j}{2}}<c<2^{\frac{j}{2}+1}$. Hence, $c \in I_{\left\lfloor\frac{j}{2}\right\rfloor} \cup I_{\left\lfloor\frac{j}{2}\right\rfloor+1}$. By construction, if $j \geq 3$, this means that the color of $c$ and $b$ are different. For $j \in{0,1,2}$ we have $b<8$ so that $c \in{2,3}$. For $c=2$, the only solutions are $(a, b, c)=(1,3,2)$ and $(2,2,2)$. The first is not monochromatic and the latter is not a valid monochromatic solution. For $c=3$, the solutions are $(2,7,3),(3,6,3)$, and $(4,5,3)$, none of which are monochromatic. Hence, the equation is not 3 -regular.

We continue by showing that for $n \geq 14$, the interval $\left[n,(10 n)^{4}\right]$ admits a monochromatic solution under any 2-coloring. To this end, let $\chi$ : $\left[n,(10 n)^{4}\right] \rightarrow{-1,1}$ be an arbitrary coloring. We use the colors $-1$ and 1 instead of the more standard 0 and 1 since we will be deriving inequalities about sums of colors and sums of $-1$ s and 1 s have more easily described properties.
Assume, for a contradiction, that $\chi$ does not admit a monochromatic solution. Since $(x, y, z)=\left(n^{2}, 80 n^{2}, 9 n\right)$ is a solution, we see that $\left[9 n, 80 n^{2}\right]$ cannot be entirely of one color. Hence, there exists
$$k \in\left[9 n, 80 n^{2}-1\right]$$
such that, without loss of generality, $\chi(k)=1$ and $\chi(k+1)=-1$.

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

Clearly $\mathbb{Z}^{+}$is closed under addition and multiplication; however, it does not contain nontrivial multiplicative or additive inverses. This is one reason that the (non-)regularity over $\mathbb{Z}^{+}$of some of the equations in the last subsection is hard to prove. If we consider fields, we will have more tools at our disposal and may be able to prove more. This does not mean the results are easy, just that we are able to bring in more algebra and number theory. We will restrict our attention to prime order fields, i.e., $\mathbb{F}_{p}$ with $p$ prime.

A word of caution: positive results over fields do not necessarily imply that the corresponding result over $Z^{+}$is true. For example, if we consider $x^{3}+y^{3}=z^{3}$, we know that there is no solution over $Z^{+}$, while it is easy to find solutions over prime-order fields $\mathbb{F}_{p}$, e.g., $3^{3}+5^{3} \equiv 4^{3}(\bmod 11)$. Note, however, if we can find that an equation is not $r$-regular over any sufficiently large prime-order fields, then we can conclude the non-regularity over $\mathbb{Z}^{+}$via the contrapositive of the Compactness Principle.

With regards to the Fermat equations, it is known [186] that for any $k \in$ $\mathbb{Z}^{+}$, for all sufficiently large primes $p$, the equation $x^{k}+y^{k}=z^{k}$ is regular over $F_{p}$, i.e., $x^{k}+y^{k} \equiv z^{k}(\bmod p)$ has a monochromatic solution with $x y z \not 0$ $(\bmod p)$ for any $r$-coloring of $\mathbb{F}_{p}$ for $p$ a sufficiently large prime. We will not prove this result here; it is shown in Section 7.1.

All results in this subsection will be presented without proof as they use methods not commonly encountered in undergraduate mathematics. We will, however, in Sections $2.5,5.2$, and 5.4, present introductions to some of the tools used in these proofs and apply these tools to other, more fundamental, results.

Theorem 2.42. For any $r \in \mathbb{Z}^{+}$, there exists a prime $P(r)$ such that the equation $x+y=u v$ is $r$-regular over $\mathbb{F}_{p}$ for all primes $p \geq P(r)$.

The above theorem is due to Sárközy [181]. After reading through Section

$2.5$, the interested reader should be able to work through its proof. Theorem $2.42$ was extended to prime power fields in [53].

As shown in the last section, $x+y=z^{2}$ is not regular – in fact, it is not even 3-regular – over $\mathbb{Z}^{+}$. The situation over $\mathbb{F}_{p}$ is much different.

Theorem 2.43. For any $r \in \mathbb{Z}^{+}$, there exists a prime $P(r)$ such that the equation $x+y=z^{2}$ is $r$-regular over $\mathbb{F}_{p}$ for all primes $p \geq P(r)$.

Theorem $2.43$ is due to Lindqvist [135], who shows more generally that $x^{j}+y^{k}=z^{\ell}$ is regular over $\mathbb{F}_{p}$ (compare this with Theorem $2.31$ and its subsequent discussion).

Lindqvist, in [135], uses techniques employed by Green and Sanders in [89] to prove the following theorem, the finite field analogue of the ${a, b, a+b, a b}$ regularity (over $\mathbb{Z}^{+}$) open question.

Theorem 2.44. For any $r \in \mathbb{Z}^{+}$, there exists a prime $P(r)$ such that the system $z=x+y, w=x y$ is $r$-regular over $\mathbb{F}{p}$ for all primes $p \geq P(r)$. Consequently, every $r$-coloring of $\mathbb{F}{p}$ admits a monochromatic set of the form ${a, b, a+b, a b}$ for all sufficiently large primes $p$.

## 数学代写|离散数学作业代写discrete mathematics代考|Hales-Jewett Theorem

We now come to a fundamental Ramsey object. In 1963 , Hales and Jewett [96] produced a new proof of van der Waerden’s Theorem by discovering a more general – and significantly stronger – result, from which van der Waerden’s Theorem follows readily. Indeed, Deuber’s Theorem (on $(m, p, c)$-sets; see Theorem 2.13) can be deduced from their result (although not easily); see [131] and also Prömel’s proof as reproduced in [93].

As the Hales-Jewett Theorem is a multi-dimensional theorem, we have need of the following standard notation.
Notation. For any set $S$, we let $S^{n}=\underbrace{S \times S \times \cdots \times S}{n \text { copies }}$. In order to state Hales and Jewett’s result, we use the following definition. (We will use the language of variable words; other authors refer to evaluated variable words as combinatorial lines, but we will not use this language.) Definition 2.45 (Variable word). A word of length $m$ over the alphabet $\mathcal{A}$ is an element of $\mathcal{A}^{m}$, which we may write as $a{1} a_{2} \ldots a_{m}$ with $a_{i} \in \mathcal{A}$ for all $1 \leq i \leq m$. Let $x$ be a variable which may take on any value in $\mathcal{A}$. A word $w(x)$ over the alphabet $\mathcal{A} \cup{x}$ is called a variable word if $x$ occurs in $w(x)$. For $i \in \mathcal{A}$, the word $w(i)$ is obtained by replacing each occurrence of $x$ with $i$.

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

I_{j}=\left{i \in \mathbb{Z}^{+}: 2^{j} \leq i \leq 2^{j+1}-1\right} \quad \text { for } \quad j=0,1,2, \ldotsI_{j}=\left{i \in \mathbb{Z}^{+}: 2^{j} \leq i \leq 2^{j+1}-1\right} \quad \text { for } \quad j=0,1,2, \ldots

ķ∈[9n,80n2−1]

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

2.5，感兴趣的读者应该能够完成它的证明。定理2.42在 [53] 中扩展到主要功率场。

Lindqvist 在 [135] 中使用 Green 和 Sanders 在 [89] 中采用的技术来证明以下定理，即一种,b,一种+b,一种b规律性（超过从+) 开放式问题。

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