Seven points in the plane such that, among any three, two are a distance $1$ apart Is there a set of seven points in the plane such that, among any three of these points, there are two, $P, R$, which are distance $1$ apart?

Yes. Normally I like to give some more motivation for an answer, but in this case it's probably hard to say anything suggestive that the illustration below does not, besides perhaps that after experimenting one might guess that the four-point "diamond" configurations are helpful (and probably necessary) in constructing such an arrangement.

(I remember seeing this problem, by the way, during a mail-in high school mathematics competition circa 1999; probably the widespread availability of the Internet makes it impractical to hold this sort of competition today.)
Edit I've posed a follow-up question, asking whether this is the unique configuration up to Euclidean motions. Servaes wrote an excellent, detailed answer showing that it is.
