Example 1 Let $\alpha, \beta(1<\alpha<\beta)$ be real numbers. Find the maximum positive integer $r$ such that: if each positive integer is arbitrarily colored with one of $r$ colors, then there always exist two positive integers $x, y$ of the same color, satisfying $\alpha \leqslant \frac{x}{y} \leqslant \beta$.
【Analysis】Assume there exists a positive integer $r$ that satisfies the problem statement.
If there exists a positive integer $N_{0}$, such that for all positive integers $n>N_{0}$, they are all of the same color, then the conclusion is obviously true. Otherwise, there exists a positive integer $t>N_{0}$, such that $t$ and $t+1$ are of different colors.
Consider $r+1$ positive integers
$$
\begin{array}{l}
a_{0}=t, a_{1}=t+1, \\
a_{i+1}=\left\lceil\alpha a_{i}\right\rceil(i=1,2, \cdots, r-1) .
\end{array}
$$

Then $a_{i+1}\sum_{k=0}^{r-1} \alpha^{k}$.
Therefore, when $\beta-\alpha^{\prime-1}>0 \Leftrightarrow 1+\log _{\alpha} \beta>r$, i.e., $r_{\text {max }}$ $=\left\lceil\log _{\alpha} \beta\right\rceil$,
$$
a_{0}>\frac{1}{\beta-\alpha^{r-1}} \sum_{k=0}^{r-1} \alpha^{k} .
$$

Choose the above positive integer $N_{0}$ sufficiently large, such that when $n>N_{0}$, we have
$$
n>\frac{1}{\beta-\alpha^{r-1}} \sum_{k=0}^{r-1} \alpha^{k} .
$$

Then $a_{r}<\alpha^{\prime-1} a_{0}+\sum_{k=0}^{r-1} \alpha^{k}<\beta a_{0}$.
Since $\left\{a_{i}\right\}$ is increasing and there are $r+1$ numbers colored with $r$ colors, there must be $0 \leqslant i<j \leqslant r$, such that $a_{i}$ and $a_{j}$ are of the same color, and
$$
\alpha a_{i} \leqslant a_{i+1} \leqslant a_{j} \leqslant a_{r}<\beta a_{0} \leqslant \beta a_{i},
$$

i.e., $\alpha \leqslant \frac{a_{j}}{a_{i}}<\beta$.
Therefore, $r=\left\lceil\log _{\alpha} \beta\right\rceil$ satisfies the condition.
Next, we prove: There exists a coloring scheme that colors each positive integer with one of $r=\left\lceil\log _{\alpha} \beta\right\rceil+1$ colors, such that there are no two numbers $x$ and $y$ of the same color, satisfying $\alpha \leqslant \frac{x}{y} \leqslant \beta$.

Let the $r$ colors be $A_{0}, A_{1}, \cdots, A_{r-1}$, and color the positive integer $n$ with $A_{i}$ if and only if $\left\llcorner\log _{a} n\right\rfloor \equiv i(\bmod r)$.
Then this coloring scheme meets the requirements.
Because for any positive integers $x$ and $y$ satisfying $\alpha \leqslant \frac{x}{y} \leqslant \beta$, we have
$$
\begin{array}{l}
1=\left\lfloor\log _{\alpha} \alpha\right\rfloor \leqslant\left\lfloor\log _{\alpha} \frac{x}{y}\right\rfloor \\
\leqslant\left\lfloor\log _{\alpha} x\right\rfloor-\left\lfloor\log _{\alpha} y\right\rfloor \leqslant\left\lceil\log _{\alpha} \frac{x}{y}\right\rceil \\
\leqslant\left\lceil\log _{\alpha} \beta\right\rceil=r-1,
\end{array}
$$

Therefore, $\left\lfloor\log _{\alpha} x\right\rfloor \neq\left\lfloor\log _{\alpha} y\right\rfloor(\bmod r)$, i.e., $x$ and $y$ are of different colors.