Extremal problems in hypergraph colourings

Abstract

Extremal problems in hypergraph colouring originate implicitly from Hilbert’s theorem on monochromatic affine cubes (1892) and van der Waerden’s theorem on monochromatic arithmetic progressions (1927). Later, with the advent and elaboration of Ramsey theory, the variety of problems related to colouring of explicitly specified hypergraphs widened rapidly. However, a systematic study of extremal problems on hypergraph colouring was initiated only in the works of Erdős and Hajnal in the 1960s. This paper is devoted to problems of finding edge-minimum hypergraphs belonging to particular classes of hypergraphs, variations of these problems, and their applications. The central problem of this kind is the Erdős–Hajnal problem of finding the minimum number of edges in an -uniform hypergraph with chromatic number at least three. The main purpose of this survey is to spotlight the progress in this area over the last several years. Bibliography: 168 titles.