A class of Ramsey-extremal hypergraphs

TL;DR

This paper characterizes all extremal 3-colorings of a 12-set hypergraph, providing insights into hypergraph Ramsey numbers and addressing open questions about size-Ramsey numbers.

Contribution

It completely classifies extremal colourings of 3-subsets in a 12-set hypergraph, advancing understanding of hypergraph Ramsey theory.

Findings

All extremal colourings of 3-subsets in a 12-set hypergraph are identified.

Properties of these extremal colourings are analyzed.

Results inform the size-Ramsey numbers of 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. Using the catalogue, we provide an answer to a question of Dudek, Fleur, Mubayi and R\"odl about the size-Ramsey numbers of hypergraphs.