Ahlswede–Daykin inequality

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 is a fundamental tool in statistical mechanics and probabilistic combinatorics (especially random graphs and the probabilistic method).

The inequality states that if ${\displaystyle f_{1},f_{2},f_{3},f_{4}}$ are nonnegative functions on a finite distributive lattice such that

${\displaystyle f_{1}(x)f_{2}(y)\leq f_{3}(x\vee y)f_{4}(x\wedge y)}$

for all x, y in the lattice, then

${\displaystyle f_{1}(X)f_{2}(Y)\leq f_{3}(X\vee Y)f_{4}(X\wedge Y)}$

for all subsets X, Y of the lattice, where

${\displaystyle f(X)=\sum _{x\in X}f(x)}$

and

${\displaystyle X\vee Y=\{x\vee y\mid x\in X,y\in Y\}}$

${\displaystyle X\wedge Y=\{x\wedge y\mid x\in X,y\in Y\}.}$

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 XYZ 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).

The story of the discovery of the Ahlswede–Daykin inequality was described in the Introduction to the A. Ahlswede et al. book:

"The history of the idea of the AD-inequality is very interesting. As Daykin came to a visit to Bielefeld, Ahlswede was just wallpapering. He stood on the ladder, and Daykin wanted to tell him from a newly proven inequality. The declaration was complicated, and Ahlswede said that probably a more general (and easier) theorem should hold. He made directly—on the ladder—a proposal which already was the AD-inequality."¹