Properly Edge-Coloured Subgraphs in Colourings of Bounded Degree

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The smallest n such that every colouring of the edges of Kn must contain a monochromatic star K1,s+1 or a properly edge-coloured Kt is denoted by f(s, t). Its existence is guaranteed by the Erd˝os-Rado Canonical Ramsey theorem and its value for large t was discussed by Alon, Jiang, Miller and Pritikin [1]. 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. 1 Properly edge-coloured subgraphs Given graphs G and H, let R(G, H) be the smallest n such that every colouring of the edges of Kn must contain a monochromatic G or a rainbow H (meaning a copy of H in which no two edges have the same colour). The Erd˝os-Rado Canonical Ramsey theorem [3] implies that R(G, H) exists if and only if either G is a star or H is acyclic. A systematic study of R(G, H) was begun by Jamison, Jiang and Ling [7] and by Chen, Schelp and Wei [2]. H...

Klas Markström, Andrew Thomason, Peter Wagner

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