Among Six People, Either Three Know Each Other or Three Are Strangers to Each Other
Represent the six people as the vertices of a regular hexagon. Draw a blue segment between a pair of vertices if the corresponding pair of people know each other and otherwise draw a red segment. A triangle with all three sides of the same color is called monochromatic; fill in such triangles with the appropriate color. There is always a monochromatic triangle! Moreover, there are always two monochromatic triangles!
See Chapter 13 of  and in particular, exercise 20, p. 300.
 M. Bóna, A Walk Through Combinatorics, 2nd. ed., Singapore: World Scientific, 2006.