A class of Ramsey-extremal hypergraphs

Abstract

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 R"odl about the size-Ramsey numbers of hypergraphs‎.