KKL theorem for the influence of a set of variables

We study the notion of the influence of a set of variables on a Boolean function, which was recently introduced by Tal. We show that for an arbitrary fixed d, every Boolean function f on n variables admits a d-set of influence at least 1/10 W≥ d(f) (log n/n)ᵈ, which is a direct generalisation of the Kahn-Kalai-Linial theorem. We give an example demonstrating essential sharpness of this result. Further, we generalise a related theorem of Oleszkiewicz regarding influences of pairs of variables.