Is this proof logically sound?

Question

Consider 17 students and three subjects: Maths, Science, and History. Any two students gossip with each other about exactly one of those three modules. Prove that there exist three students gossiping with each other about the same module.

If possible could someone validate this proof:

Since each student gossips with one other student, there are ceiling(17/2) = 8 separate conversations.

Since there are 3 subjects ceiling(8/3) = 3

So by the pigeonhole principle, there must exist three students gossiping about the same subjects. Is this ok?

Any tips would be appreciated!

Answer 1

This problem can be solved in two steps.

1. First, let us prove that if we have only two subjects (maths and science) and six students and any two students gossip with each other about exactly one of those two subjects. Prove that there exist three students gossiping with each other about the same two subject.

Solution. Take an arbitrary student, say John. He gossips with five others about our two subjects. By the pigeonhole principle, there are at least three students with whom John gossips about the same subject, say maths.

If those three students gossip with each other about science, then the problem is solved.

If at least two of those three are gossiping about maths, add John to that couple and we've solved the problem again.

2. Your case with three subjects and 17 students reduces to case 1 in the same way. Try it on your own.