6. Each point on a circle is colored one of three colors: red, yellow, or blue, and all three colors appear. Now, $n$ points are chosen from the circle. If among these points, there always exist three points that form a triangle with vertices of the same color and an obtuse angle, then the minimum possible value of $n$ is $\qquad$ .
6.13.

First, we state that \( n \geqslant 13 \).
If \( n \geqslant 13 \), then by the pigeonhole principle, among these \( n \) points, there must be \(\left[\frac{13-1}{3}\right]+1=5\) points of the same color (let's assume they are red).
Take a diameter, whose endpoints do not belong to these five points.
By the pigeonhole principle, among these five red points, there are always three on the same side of this diameter, and the triangle formed by these three points must be an acute triangle.

When \( n=12 \), consider the following scenario: select three inscribed squares \( A_{1} A_{2} A_{3} A_{4} \), square \( B_{1} B_{2} B_{3} B_{4} \), and square \( C_{1} C_{2} C_{3} C_{4} \). The vertices of these three squares are respectively red, yellow, and blue. In this case, all triangles with three vertices of the same color are right triangles. Therefore, \( n=12 \) does not meet the requirement.
In summary, the smallest possible value of \( n \) is 13.