Friedgut–Kahn–Kalai–Keller antipodal correlation conjecture

Prove the Friedgut–Kahn–Kalai–Keller conjecture for increasing Boolean functions on the discrete hypercube: For all n ≥ 1 and all increasing functions f,g:{0,1}^n→{0,1} such that g is antipodal (i.e., g(x)=1−g(1−x) for all x), establish the lower bound E[fg]−E[f]E[g] ≥ (1/4)·min_i Inf_i[f], where Inf_i[f] denotes the (uniform) influence of coordinate i on f.

Background

The paper studies quantitative lower bounds on the covariance of increasing functions on the hypercube in terms of their influences, strengthening Talagrand-type inequalities under supermodularity or submodularity. A classical reduction considers an antipodal function g, which is closely linked to extremal set theory.

Friedgut, Kahn, Kalai, and Keller showed that establishing the stated correlation bound when one function is antipodal is equivalent to Chvátal’s conjecture. The present paper confirms this bound within a structured regime (super/submodularity), but the conjecture remains open in full generality.

References

Friedgut, Kahn, Kalai, and Keller showed that the following conjecture is equivalent to the celebrated Chv\ atal conjecture: If $f,g:{0,1}n\to{0,1}$ are increasing and $g$ is antipodal, then \begin{equation}\label{ineq:Chvatal} \mathbb{E}[fg]-\mathbb{E}[f]\mathbb{E}[g] \ge\tfrac{1}{4}\cdot \min_i \mathrm{Inf}_i[f]. \end{equation}

ineq:Chvatal:

E[fg]−E[f]E[g]≥14⋅min⁡iInfi[f].\mathbb{E}[fg]-\mathbb{E}[f]\mathbb{E}[g] \ge\tfrac{1}{4}\cdot \min_i \mathrm{Inf}_i[f].

— Talagrand-Type Correlation Inequalities for Supermodular and Submodular Functions on the Hypercube  (2510.22307 - Chang et al., 25 Oct 2025) in Conjecture 1.1, Related Work (Section 1.4)