For any five points in $S$, coloring them red or blue must result in three points of the same color (pigeonhole principle), so conclusion (1) holds. There are finitely many triangles with three vertices of the same color, and among them, there must be one with the smallest area (let it be $\triangle A B C$). Then $\triangle A B C$ satisfies conclusion (2). If not, each side of $\triangle A B C$ would have a point of a different color, and these three different-colored points would form another triangle with three vertices of the same color, which is smaller than $\triangle A B C$, leading to a contradiction.