1. ## Pigeon-hole Principle (3)

One afternoon, a mathematics library had several visitors. A librarian noticed that it was impossible to find three visitors so that no two of them had met in the library that afternoon. Prove that then it was possible to find two moments of time that afternoon so that each visitor was in the library at one of those two moments.

2. ## Re: Pigeon-hole Principle (3)

I'm not really understanding the question. I'm imagining a counterexample where people come in two at a time, meet, and leave, with people coming in enough times so that every 3-subset of the set of all visitors contains at least one pair of people that have met. For example, with 5 people,

{1,2}, {1,4}, {1,3}, {2,4}, {2,5}, {3,5}.

What am I doing wrong?

3. ## Re: Pigeon-hole Principle (3)

Originally Posted by topspin1617
I'm not really understanding the question. I'm imagining a counterexample where people come in two at a time, meet, and leave, with people coming in enough times so that every 3-subset of the set of all visitors contains at least one pair of people that have met. For example, with 5 people,

{1,2}, {1,4}, {1,3}, {2,4}, {2,5}, {3,5}.

What am I doing wrong?
You have a point.

Edit: A poster on another forum said "The problem implicitly assumes that each visitor spent only one contiguous lapse of time in the library - otherwise we may have each pair of visitors being together at some point in the library, without more than two visitors being simultaneously in the library at any moment in time."