Is there any known improvement on the Kahn-Kalai-Linial inequality (on the influences of boolean functions) in the special case in which $f$ is the indicator function of an *intersecting* monotonic set system? More concretely, is there an absolute constant $C>0$ such that the following statement holds:

If $f:\mathcal{P}([n]) \to \lbrace 0,1\rbrace$ is the indicator of an intersecting upset, then there exists $x\in [n]$ such that $I_x(f)\geq C/\sqrt{n}$.

(Note that the "tribes" example of a half-sized system in which all influences are $\ll \log n /n$ is certainly not intersecting.)

Background: The $x$th influence of a boolean function $f$ is defined as

$$I_x(f) = \mathbf{E}(f(X)\neq f(X \Delta \lbrace x\rbrace),$$

($\Delta$ is symmetric difference) where $X$ is drawn randomly and uniformly from $\mathcal{P}([n])$. In particular, if $f$ is the indicator of a monotonic set system $\mathcal{U}\subset\mathcal{P}([n])$ (monotonic meaning $X\subset Y$ and $X\in\mathcal{U}$ implies $Y\in\mathcal{U}$), $I_x(f)$ is the number of sets $X\in \mathcal{U}$ containing $x$, minus the number of such sets not containing $x$, divided by $2^{n-1}$.

intersecting set system? – Seva Aug 21 '12 at 9:41