This problem can be approached using Ramsey theory, specifically Ramsey's theorem for two colors, which states that in any complete graph of $R(3, 3)$ (the Ramsey number for $3$-cliques), there must be either a monochromatic triangle of one color or a monochromatic triangle of another color.

Here's a proof by considering the setup and applying graph theory.

1. Represent the People as a Graph:

   • Consider each of the 6 people as vertices of a graph.
   • Draw an edge between two vertices if they are acquainted. If they are not acquainted, there is no edge between them (or alternatively, you can represent lack of acquaintance as a different color edge in a two-colored complete graph).

2. Application of Ramsey's Theorem:

   • For 6 vertices, Ramsey’s theorem states that in any two-coloring of a complete graph $K_6$, where each edge is colored either blue or red (representing acquaintance or non-acquaintance), there must exist a monochromatic triangle.
   • This implies that no matter how the edges are colored (whether representing acquaintance or non-acquaintance), there must be a triangle where all edges are the same color.

3. Interpreting the Monochromatic Triangle:

   • A monochromatic triangle in the context of this problem means either:
     • Three people who are all mutually acquainted (if all edges in the triangle are red, say), or
     • Three people who are all mutually non-acquainted (if all edges in the triangle are blue).
   • Thus, Ramsey’s theorem guarantees that we will have either a fully acquainted trio or a fully non-acquainted trio among any 6 people.

4. Conclusion:

   • Therefore, in any group of 6 people, there must be 3 who are all mutually acquainted or 3 who are all mutually non-acquainted, as guaranteed by the existence of a monochromatic triangle in $K_6$.

This proves that in a group of 6 people, we are guaranteed to find 3 people who are either all acquainted or all not acquainted.

Would you like more details on Ramsey theory, or do you have any questions?

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Here are 5 related questions to expand understanding:

1. What is Ramsey's theorem, and how does it generalize to larger groups?
2. How is the concept of a monochromatic triangle used in combinatorial proofs?
3. Can we use this theorem to find groups of 4 people who are all acquainted or all non-acquainted?
4. What are other applications of Ramsey numbers in graph theory?
5. How would the proof differ if we had only 5 people instead of 6?

Tip: Ramsey theory often deals with "complete disorder" and shows that even in apparently random structures, certain patterns must emerge.