A class of Ramsey-extremal hypergraphs

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. We also provide an answer to a question of Dudek, La Fleur, Mubayi and Rodl about the size-Ramsey numbers of hypergraphs.