On a Correlation Inequality of Farr

Colin McDiarmid
1992 Combinatorics, probability & computing  
Suppose that each vertex of a graph independently chooses a colour uniformly from the set {1, . . . , k}; and let S i be the random set of vertices coloured i. Farr shows that the probability that each set S i is stable (so that the colouring is proper) is at most the product of the k probabilities that the sets S i separately are stable. We give here a simple proof of an extension of this result.
doi:10.1017/s096354830000016x fatcat:u4ddduuaabf7xchzh6hzosjzbi