The AhlswedeâÂÂDaykin inequality , also known as the four functions theorem (or inequality), is a correlation-type inequality for four functions on a finite distributive lattice. It is a fundamental tool in statistical mechanics and probabilistic combinatorics (especially random graphs and the probabilistic method).
The inequality states that if are nonnegative functions on a finite distributive lattice such that
for all x, y in the lattice, then
for all subsets X, Y of the lattice, where
and
The AhlswedeâÂÂDaykin inequality can be used to provide a short proof of both the Holley inequality and the FKG inequality. It also implies the XYZ inequality.
For a proof, see the original article or .
The "four functions theorem" was independently generalized to 2k functions in and .