5. Given 9 points in space, where no four points are coplanar, line segments are drawn between every pair of points. These line segments can be colored blue or red, or left uncolored. Find the minimum value of $n$ such that, by arbitrarily coloring any $n$ of these line segments with one of the two colors (red or blue), the set of these $n$ line segments will necessarily contain a triangle with all sides of the same color.
5. The minimum value of $n$ is 33.

Lemma: For the edges of a complete graph of order 5, there exists a coloring method such that the colored graph contains no monochromatic triangles.
As shown in Figure (1), solid and dashed lines represent edges of two different colors, and at this point, there are no monochromatic triangles in the graph.
Proof of the original problem: As shown in Figure (2), the 9 vertices are numbered $1,2,3, \ldots, 9$. Among them, vertices $\{1,3\},\{1,4\},\{2,3\},\{2,4\}$ are all connected by edges, and vertices 1 and 2 within the circle are not connected; similarly, vertices in different circles are all connected, while vertices 3 and 4, 5 and 6, 7 and 8 within the same circle are not connected. By the lemma, this graph contains no monochromatic triangles. It is easy to calculate that the number of edges in this graph is $C\}-4=32$, so $n \geqslant 33$.

On the other hand, if the number of colored segments is at least 33, since there are $C\}=36$ segments in total, the number of uncolored segments is at most 3. If vertex $A_{1}$ is connected by an uncolored segment, then remove $A_{1}$ and all segments connected to it; if there is still a vertex $A_{2}$ connected by an uncolored segment in the remaining graph, then remove $A_{2}$ and all segments connected to it, and so on. Since there are at most 3 uncolored segments, at most 3 vertices and the segments connected to them can be removed, leaving at least 6 vertices, and every pair of these vertices is connected by a segment colored red or blue. By the conclusion of Example 1 in this lecture, there must be a monochromatic triangle.
In summary, the minimum value of $n$ is 33.