Six people: three friends or three strangers

1. Colour each pair of the six people red (friends) or blue (strangers). Fix a person $v$. Among the $5$ pairs containing $v$, at least $3$ have the same colour; say red, towards $a,b,c$.
- If one of the pairs $ab$, $bc$, $ca$ is red, say $ab$, then $v,a,b$ are pairwise friends.
- Otherwise $ab$, $bc$, $ca$ are all blue and $a,b,c$ are pairwise strangers.

The case of three blue pairs at $v$ is symmetric.

2. Put $5$ people on a regular pentagon: two people are friends if they are adjacent on the pentagon, strangers otherwise. Friends form the $5$-cycle and strangers the "pentagram" cycle. Neither graph contains a triangle.

Lesson. A pigeonhole argument at one vertex, then a case analysis; extremal sharpness comes from a symmetric colouring.