Properly Edge-Coloured Subgraphs in Colourings of Bounded Degree

Abstract

The smallest n such that every colouring of the edges of K n must contain a monochromatic star K 1,s+1 or a properly edge-coloured K t is denoted by f (s, t). Its existence is guaranteed by the Erdős–Rado Canonical Ramsey theorem and its value for large t was discussed by Alon, Jiang, Miller and Pritikin (Random Struct. Algorithms 23:409–433, 2003). In this note we primarily consider small values of t. We give the exact value of f (s, 3) for all s ≥ 1 and the exact value of f (2, 4), as well as reducing the known upper bounds for f (s, 4) and f (s, t) in general.