Ahlswede–Daykin inequality

Remarks

It is also called the four functions theorem. Every finite distributive lattice embeds in the lattice of subsets of a finite set, so it suffices to treat the power set of $\set{1, \ldots, n}$. The proof is by induction on $n$; the case $n = 1$ is a direct check of an inequality between four products of nonnegative numbers.

Example: with all four functions equal to $1$, the hypothesis holds and the conclusion is $|A| \, |B| \le |A \vee B| \, |A \wedge B|$ for any subsets $A, B$ of $L$. Taking $f_1 = f_2 = f_3 = f_4 = \mu$ for a function $\mu$ with $\mu(x)\mu(y) \le \mu(x \vee y)\mu(x \wedge y)$ gives the FKG inequality and, as a special case, the Harris inequality.

Going deeper

For a student

The four functions theorem is the common source of the correlation inequalities of combinatorics. It contains the FKG inequality, the Harris inequality and Kleitman's lemma on monotone families of sets. These are used for random graphs, percolation and the statistical mechanics of spin systems, where increasing events tend to occur together.