4. Given a positive integer $N \geqslant 3$. If a set of $N$ points, each either red or blue, on the coordinate plane has each point with a unique x-coordinate, then the set of points is called "acceptable". $P(x)$ is a polynomial, and $A$ is an acceptable set. If the upper side of the graph of $P(x)$ contains no red points and the lower side contains no blue points, or the upper side contains no blue points and the lower side contains no red points, then $P(x)$ is said to "distinguish" the set $A$. It is known that any acceptable set can be distinguished by a polynomial of degree not exceeding $k$. Find the minimum value of $k$.


4. $k_{\min }=N-2$.

First, we point out that $k=N-2$ satisfies the condition.
Choose any $N-1$ points from the set $A$, and construct a polynomial of degree no more than $N-2$ such that its graph passes through these points. Clearly, this polynomial distinguishes $A$.

Next, we show that a polynomial of degree less than $N-2$ cannot distinguish the set $A$.

Let $f$ be a polynomial of degree $N-2$, and take $N$ points on its graph with alternating colors.

Assume there exists a polynomial $P$ of degree no more than $N-3$ that distinguishes these points, without loss of generality, assume there are no red points below its graph and no blue points above its graph.
Consider the polynomial of degree $N-2$
$$
Q(x)=f(x)-P(x) \text {. }
$$

If $r$ and $b$ represent the x-coordinates of any given red and blue points, respectively, then
$$
P(r) \leqslant f(r), P(b) \geqslant f(b) .
$$

Thus, $Q(r) \geqslant 0, Q(b) \leqslant 0$.
Note that, if there exists $s0$.

Therefore, in any interval formed by a red point on the left and a blue point on the right, there is a point where $Q^{\prime}(x)$ is negative, and in any interval formed by a blue point on the left and a red point on the right, there is a point where $Q^{\prime}(x)$ is positive.

This indicates that $Q^{\prime}(x)$ changes sign at least $N-2$ times, meaning $Q^{\prime}(x)$ has at least $N-2$ real roots. This contradicts the fact that $Q^{\prime}(x)$ is a polynomial of degree $N-3$.