  In 1991, McKay and Radziszowski proved that, however each 3-subset of a
13-set is assigned one of two colours, there is some 4-subset whose four
3-subsets have the same colour. More than 25 years later, this remains the only
non-trivial classical Ramsey number known for hypergraphs. In this article, we
find all the extremal colourings of the 3-subsets of a 12-set and list some of
their properties. Using the catalogue, we provide an answer to a question of
Dudek, Fleur, Mubayi and R\"odl about the size-Ramsey numbers of hypergraphs.
