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On a Correlation Inequality of Farr

1992
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Combinatorics, probability & computing
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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