Suppose the edges of the complete graph on 17 vertices are coloured in three colours. It is shown that at least five monochromatic triangles must arise. INTRODUCTION By a proper colouring of a graph G in n colours is meant a way of assigning n or fewer colours to the edges of G in such a way that no monochromatic triangle results. We are interested in edge-colourings of complete graphs. If K r denotes the complete graph on r vertices, then Ramsey's theorem guarantees the existence of a number r

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... n (2) such that K r has a proper colouring in n colours if and only if r < r n (2). (For a discussion see [7] .) These numbers have been investigated; however only r 2 {2) and r 3 (2) are known. PROPOSITION 1. [6] r 2 (2) = 6; r 3 (2) = 17. Since T"2(2) = 6, any colouring of K § must contain a monochromatic triangle. In fact, it can be shown that at least two triangles must occur. This follows from a more general result of Goodman: PROPOSITION 2. [1] Iii any edge-colouring of K r in two colours there must be at least f(r) monochromatic triangles, where: f{r) --r(r -2)(r -4) if r is even; f( r ) = ^r(r -l)(r -5) if r = 1 {mod 4); /(»•) = ^( r + !)(»• -3 )( r -4) if r = 3 (mod 4). Moreover, these bounds can all be attained. When the number of colours is 3, there must clearly be a function like / , but we know very little about it.-A recent paper by Goodman [2] discusses the 3-colour