### Monochromatic triangles in three colours

S.S. Sane, W.D. Wallis
1988 Bulletin of the Australian Mathematical Society
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
more » ... tence of a number r 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