Problem 1450

1. Define the Problem in Graph Theory Terms:
We start by considering a complete graph $G$ with vertices $v_1, v_2, \ldots, v_{n+1}$. For each pair of vertices $v_i$ and $v_j$ where $1 \leq i < j \leq n+1$, we color the edge between $v_i$ and $v_j$ with the color assigned to the subset $\{i, i+1, \ldots, j-1\}$.

2. Ramsey Number and Monochromatic Triangles:
By Ramsey's theorem, for any given number of colors $c$, there exists a minimum number $R(c)$ such that any complete graph with at least $R(c)$ vertices will contain a monochromatic triangle. Here, we are given $1391$ colors, so we need $n$ to be large enough such that $n > R_{1391}$.

3. Existence of Monochromatic Triangle:
If $n > R_{1391}$, then the complete graph $G$ with $n+1$ vertices will contain a monochromatic triangle. Let this triangle be formed by the vertices $v_x, v_y, v_z$ with $x < y < z$.

4. Identify the Subsets:
The edges of this monochromatic triangle correspond to the following subsets:
- The edge $v_x v_y$ corresponds to the subset $A = \{x, x+1, \ldots, y-1\}$.
- The edge $v_y v_z$ corresponds to the subset $B = \{y, y+1, \ldots, z-1\}$.
- The edge $v_x v_z$ corresponds to the subset $A \cup B = \{x, x+1, \ldots, z-1\}$.

5. Conclusion:
Since the triangle $v_x v_y v_z$ is monochromatic, the subsets $A$, $B$, and $A \cup B$ are all of the same color.

$\blacksquare$