Six people form a club. In the club, two people either like each other or dislike each other. Show that either there is a group of three all of whom like each other OR there is a group of three all of whom dislike each other.

## Solution

For convenience put the six people around a circle as in the figure below. Connect people who like each other. There are two classes of person: people that like more than they dislike, that is, they like at least three others. Call these people class $\mathcal{L}$. There are also possibly people who dislike more than they like. Call these people class $\mathcal{D}$.

Pick one person, say A who is in class $\mathcal{L}$. This means A likes at least three others. Say that these are B, C, and D, as in the figure. Now if at least one other pair among B, C, D like each other then right there there is a group (including A) of three people who mutually like each other. And if none of B, C, and D like each other (as is the case in the figure), then there is a group of three people who dislike each other. That proves what we want, if A happens to be in class $\mathcal{L}$.

Complete the argument in case A happens to be in class $\mathcal{D}$.