To prove this statement, we can use the pigeonhole principle. The statement essentially requires us to show that for a group of mathematicians, there exists either:

1. A set of six mathematicians, none of whom were asleep at the same time, or
2. A set of six mathematicians who were all asleep simultaneously at some point in time.

This is a variation of Ramsey Theory applied to two types of configurations within a group. Specifically, this is related to Ramsey Number $R(6, 6) = 36$, which means that any graph with 36 vertices will contain either a complete subgraph $K_6$ or an independent set $I_6$. Here’s the detailed proof:

Proof:

1. Interpretation in Graph Theory:

   • Consider each mathematician as a vertex in a graph.
   • Draw an edge between two vertices if the two mathematicians were not asleep at the same time.
   • No edge between two vertices means that these two mathematicians were asleep simultaneously at some point.
   • We are asked to find either a set of 6 vertices where all pairs are connected (a $K_6$, where no two were asleep at the same time) or a set of 6 vertices with no connections between them (an independent set $I_6$, where all were asleep at the same time).

2. Using the Pigeonhole Principle with Ramsey Numbers:

   • For any two-colored complete graph with 36 vertices, the Ramsey number $R(6, 6) = 36$ tells us that there will always be either a $K_6$ or an $I_6$ subgraph.
   • This means, within a group of 36 mathematicians, there must either exist a set of 6 who were never asleep at the same time (no edge between them, forming a $K_6$) or a set of 6 mathematicians who were all asleep at some point (an $I_6$).

3. Conclusion:

   • Since the minimum required number of mathematicians to guarantee either situation is 36, any group of 36 mathematicians will meet the conditions of either a $K_6$ or an $I_6$.
   • Therefore, within any group of 36 mathematicians, there will necessarily exist either:
     • A group of 6 mathematicians who were never asleep at the same time, or
     • A group of 6 mathematicians who were all asleep at the same time.

This satisfies the conditions of the statement, thus proving that such a configuration always exists for a group of 36 mathematicians.

Would you like more details on any specific part of this proof?

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Related Questions:

1. What is the pigeonhole principle, and how is it used in combinatorics?
2. How does Ramsey Theory apply to real-world scenarios?
3. What are other applications of Ramsey numbers beyond graph theory?
4. Can the proof above be generalized for other numbers, like proving $R(5,5)$?
5. How does the concept of independent sets work in graph theory?

Tip: Understanding the pigeonhole principle can simplify complex proofs, especially in combinatorics and probability.