Graphs containing triangles are not 3-common

A finite graph G is {\em k-common} if the minimum (over all k-colourings of the edges of K_n) of the number of monochromatic labelled copies of G is asymptotically equal, as n tends to infinity, to the expected number of such copies in a random k-colouring of the edges of K_n. Jagger, \u{S}\u{t}oví\u{c}ek and Thomason showed that graphs which contain K₄ are not 2-common. We prove that graphs which contain K₃ are not 3-common.