Ahlswede–Daykin inequality

A fundamental tool in statistical mechanics and probabilistic combinatorics (especially random graphs and the probabilistic method), the Ahlswede–Daykin inequality (Ahlswede & Daykin 1978), also known as the four functions theorem (or inequality), is a correlation-type inequality for four functions on a finite distributive lattice.

It states that if ${\displaystyle f_{1},f_{2},f_{3},f_{4}}$ 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 Ahlswede–Daykin inequality can be used to provide a short proof of both the Holley inequality and the FKG inequality. It also implies the Fishburn–Shepp inequality.

For a proof, see the original article (Ahlswede & Daykin 1978) or (Alon & Spencer 2000).

The "four functions theorem" was independently generalized to 2k functions in (Aharoni & Keich 1996) and (Rinott & Saks 1991).

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