On a Correlation Inequality of Farr

Suppose that each vertex of a graph independently chooses a colour uniformly from the set {1, . . . , k}; and let Si be the random set of vertices coloured i. Farr shows that the probability that each set Si is stable (so that the colouring is proper) is at most the product of the k probabilities that the sets Si separately are stable. We give here a simple proof of an extension of this result.