Graph theory question involving people seated at a round table

Question

50 mathematicians attend a conference at which each knows 25 other attendees. Show that you can select 4 of them who can then be seated at a round table, such that each person at the table knows the two people he or she is sitting next to.

I would like a hint please. I don't know how to get started.

Answer 1

Hint:

Select any vertex $u$ and any two of its neighbors $v$ and $w$. If we could find a common neighbor of $v$ and $w$ (other than $u$), then we would have a 4-cycle. Why do the high degrees of $u$ and $v$ guarantee that such a vertex exists?

Answer 2

HINT: Start with two mathematicians who don’t know each other. Show that they must have at least $2$ acquaintances in common.

Answer 3

Select two people, a and c, who don't know each other. The number of people who know a, plus the number of people who know c, is 50. However there are only 48 people in the group who are not a or c, so...