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

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

The simplest of nontrivial equations is probably $x+y=z$ (clearly, $x=y$ has the Ramsey property, while it is easy to show that $y=x+b, b \neq 0$, and $y=c x, c \neq 0,1$, do not). By Corollary $2.2$, the existence of $\widehat{w}(2,1 ; r)$ shows

that we do have a monochromatic solution to $x+y=z$ with $x=a$ and $y=d$. However, such a monochromatic solution was proven by Schur [186] before van der Waerden’s Theorem was proven. We follow Schur’s original proof.
Theorem 2.15 (Schur’s Theorem). Let $r \in \mathbb{Z}^{+}$. There exists a minimal positive integer $s=s(r)$ such that every $r$-coloring of $[1, s]$ contains a monochromatic solution to $x+y=z$.

Proof. We will show that $s(r)<$ er!, thereby proving existence. Let $n_{0} \in$ $\mathbb{Z}^{+}$be arbitrary and assume that we have an $r$-coloring of $\left[1, n_{0}\right]$ with no monochromatic solution to $x+y=z$. Some color occurs at least $\frac{n_{0}}{r}$ times. Let this be color 1 and let $x_{1}, x_{2}, \ldots, x_{n_{1}}$ be all of the integers of color 1 , with $n_{1} \geq \frac{n_{0}}{r}$. By assumption,
$$S_{1}=\left{x_{k}-x_{1}: 2 \leq k \leq n_{1}\right}$$
is void of color $1 .$
Within $S_{1}$, some color occurs at least $\frac{n_{1}-1}{r-1}$ times. Let this be color 2 (it cannot be color 1). Let $y_{1}, y_{2}, \ldots, y_{n_{2}}$, with $n_{2} \geq \frac{n_{1}-1}{r-1}$, be all of the integers of color 2. By assumption,
$$S_{2}=\left{y_{k}-y_{1}: 2 \leq k \leq n_{2}\right}$$
is void of color 2. Since $y_{j}=x_{k_{j}}-x_{1}$ we see that $y_{i}-y_{1}=x_{k_{i}}-x_{k_{1}}$ so that $x_{k_{1}}+\left(y_{i}-y_{1}\right)=x_{k_{i}}$. Hence, $S_{2}$ must also be void of color 1 .

We continue in this manner to define sets $S_{i}$ of size $n_{i}-1$, where $n_{i} \geq$ $\frac{n_{i-1}-1}{r-i+1}$, that are void of colors $1,2, \ldots, i$. Hence, we have $n_{0} \leq r n_{1}$ and $n_{i} \leq$ $(r-i+1) n_{i+1}+1$ for $1 \leq i \leq k$ for some $k<r$ (since $S_{r}$ must be void of all colors).
Stringing these together, we have
\begin{aligned} n_{0} \leq r n_{1} \leq r\left((r-1) n_{2}+1\right) &=r+r(r-1) n_{2} \ & \leq r+r(r-1)+r(r-1)(r-2) n_{3} \ & \vdots \ & \leq \sum_{i=0}^{r} \frac{r !}{i !} \ &<r ! \sum_{i=0}^{\infty} \frac{1}{i !}=e r ! \end{aligned}
thereby proving the theorem.

Thus far we have seen that the equations $x+y=z$ and $x+y=2 z$ each admit monochromatic solutions under any finite coloring of $\mathbb{Z}^{+}$. On the other hand, it is easy to see that by coloring the entire intervals $\left[2^{i}, 2^{i+1}-1\right]$ with color $i$ modulo 2 , we have no guaranteed monochromatic solution to $x=2 y$ even for 2 colors. In order to discuss these types of results, we make the following definition.

Definition $2.19$ ( $r$-regular, regular). Let $r \in \mathbb{Z}^{+}$. We say that an equation $\mathcal{E}$ is $r$-regular if every $r$-coloring of $\mathbb{Z}^{+}$admits a monochromatic solution to $\mathcal{E}$. If $\mathcal{E}$ is $r$-regular for all $r \in \mathbb{Z}^{+}$, then we say that $\mathcal{E}$ is regular.

In this subsection we will limit our attention to linear homogeneous equations. We start with a lesser-known result of Rado [163], who was a student of Schur.

Theorem 2.20. Let $n \in \mathbb{Z}^{+}$with $n \geq 3$ and let $c_{i}, 1 \leq i \leq n$, be non-zero integers. The equation $\sum_{i=1}^{n} c_{i} x_{i}=0$ is 2 -regular if for some $i_{1}$ and $i_{2}$ we have $c_{i_{1}}>0$ and $c_{i_{2}}<0$.

Proof. By equating variables as necessary, it suffices to prove that every 2coloring of $\mathbb{Z}^{+}$admits a monochromatic solution to $a x+b y=c z$ for any $a, b, c \in \mathbb{Z}^{+}$.

Assume, for a contradiction, that there exists a 2-coloring of $\mathbb{Z}^{+}$with no monochromatic solution. We may assume that every dilation of $\mathbb{Z}^{+}$, say $d \mathbb{Z}^{+}$, contains elements of both colors; otherwise, $(d b c, d a c, d a b)$ would be a monochromatic solution and we would be done.

Let $a c$ be red and let $a c j$ be blue $(j \geq 2)$ as both are members of the dilation $a c \mathbb{Z}^{+}$. Within $b c(j-1) \mathbb{Z}^{+}$there exists a blue element. Let this element be $t_{1}$ and let
$$t_{1}=b c(j-1) k$$

Consider
$$u_{1}=a b(j-1) k+a b j .$$
Since $\left(t_{1}, a c j, u_{1}\right)$ is a solution, we may assume that $u_{1}$ is red. Now consider
$$t_{2}=b c(j-1)(k+1)$$
Since $\left(t_{2}, a c, u_{1}\right)$ is a solution, we may assume that $t_{2}$ is blue.
We proceed in this fashion (starting with $\left.u_{2}=a b(j-1)(k+1)+a b j\right)$ to deduce that
$$t_{i}=b c(j-1)(k+i-1)$$
is blue for all $i \in \mathbb{Z}^{+}$. Since $\left{t_{i}\right}_{i \in \mathbb{Z}}+\supseteq b c(j-1) k \mathbb{Z}^{+}$, we have a monochromatic dilation of $\mathbb{Z}^{+}$, contradicting our assumption.

Before moving on to other equations, we give some results about Rado numbers. Unlike van der Waerden and Schur numbers, Rado numbers are a class

of numbers covering all systems of equations (typically linear and homogeneous). No over-arching formula is known for arbitrary systems (or even an arbitrary equation). To date, the determination of these numbers has almost solely been restricted to single equations. Thus, we will restrict our definition to single equations.

Definition 2.27 (Rado numbers). Let $r \in \mathbb{Z}^{+}$. Let $\mathcal{E}$ be an $r$-regular linear homogeneous equation. The minimal positive integer $n=n(\mathcal{E} ; r)$ such that every $r$-coloring of $[1, n]$ admits a monochromatic solution to $\mathcal{E}$ is called the $r$-color Rado number for $\mathcal{E}$.

We gather, in Table $2.3$, all known completely determined 2-color Rado numbers (as of this writing). As the proofs of these numbers are mostly elementary (but, perhaps, clever) color-forcing arguments, we will not provide proofs. The reader may find proofs by following the reference(s) given by each result. In Table $2.3$, parameters $a, a_{i}, b, k$, and $\ell$ are positive integers.

Note that when restricted to two colors, Theorem $2.20$ states that we need not satisfy the criteria of Theorem $2.22$ (a subset of coefficients summing to 0 ) in order for the 2-color Rado number to exist.

Although there are known numerical values for more than two colors for Rado numbers for some specific equations, there are no formulas as we have for two colors, except for a few specific cases.

As we can see from Table $2.3$, we are homing in on the determination of the 2-color Rado number for an arbitrary homogeneous linear equation.

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

S_{1}=\left{x_{k}-x_{1}: 2 \leq k \leq n_{1}\right}S_{1}=\left{x_{k}-x_{1}: 2 \leq k \leq n_{1}\right}

S_{2}=\left{y_{k}-y_{1}: 2 \leq k \leq n_{2}\right}S_{2}=\left{y_{k}-y_{1}: 2 \leq k \leq n_{2}\right}

n0≤rn1≤r((r−1)n2+1)=r+r(r−1)n2 ≤r+r(r−1)+r(r−1)(r−2)n3 ⋮ ≤∑一世=0rr!一世! <r!∑一世=0∞1一世!=和r!

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## MATLAB代写

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