1. Let $S$ be a set of $n(n \geqslant 5)$ points in the plane. If any four points chosen from $S$ have at least one point connected to the other three, then which of the following conclusions is correct? $\qquad$
(1) There is no point in $S$ that is connected to all other points;
(2) There is at least one point in $S$ that is connected to all other points;
(3) There are at most two points in $S$ that are not connected to all other points;
(4) There are at most two points in $S$ that are connected to all other points.
-、1. (2).
In the point set $S$, all points are connected to each other, which clearly satisfies the problem. Therefore, conclusions (1) and (4) are incorrect.

Suppose $A$, $B$, and $C$ are three points in the point set $S$ that are not connected to each other, but the remaining $n-3$ points are all connected to each other. This also clearly satisfies the problem. Hence, conclusion (3) is also incorrect.

If all points in the point set $S$ are connected to each other, then conclusion (2) is obviously true; otherwise, assume points $P$ and $Q$ are not connected. Take any two other points in the point set $S$, say $R$ and $T$. Among the four points $P$, $Q$, $R$, and $T$, there must be one point that is connected to the other three. This point cannot be $P$ or $Q$, so it must be one of $R$ or $T$ (let's say $R$). By the arbitrariness of points $R$ and $T$, we know that there is at least one point in the point set $S$ that is connected to all other points. Therefore, conclusion (2) is correct.