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 \( f_1,f_2,f_3,f_4 \) are nonnegative functions on a finite distributive lattice such that

\( f_1(x)f_2(y)\le f_3(x\vee y)f_4(x\wedge y) \)

for all x, y in the lattice, then

\( f_1(X)f_2(Y)\le f_3(X\vee Y)f_4(X\wedge Y) \)

for all subsets X, Y of the lattice, where

\( f(X) = \sum_{x\in X}f(x) \)

and

\( X\vee Y = \{x\vee y\mid x\in X, y\in Y\} \)
\( X\wedge Y = \{x\wedge y\mid x\in X, y\in Y\}. \)

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 Fishburn–Shepp inequality.

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

Generalizations

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

References

Ahlswede, Rudolf; Daykin, David E. (1978), "An inequality for the weights of two families of sets, their unions and intersections", Probability Theory and Related Fields 43 (3): 183–185, doi:10.1007/BF00536201, ISSN 0178-8051, MR 0491189
Alon, N.; Spencer, J. H. (2000), The probabilistic method. Second edition. With an appendix on the life and work of Paul Erdős., Wiley-Interscience, New York, ISBN 0-471-37046-0, MR 1885388
Fishburn, P.C. (2001), "a/a110440", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Aharoni, Ron; Keich, Uri (1996), "A Generalization of the Ahlswede Daykin Inequality", Discrete Mathematics 152: 1–12, doi:10.1016/0012-365X(94)00294-S
Rinott, Yosef; Saks, Michael (1991), "Correlation inequalities and a conjecture for permanents", COMBINATORICA 13 (3): 269–277, doi:10.1007/BF01202353