Kalai–Meshulam's fractional Helly conjecture for polynomial homological shatter functions

Let $d$ be a nonnegative integer, let $k$ be an integer with $0\leq k\leq d$, and let $A>0$. For a set system $\mathcal{F}$ in $\mathbb{R}^d$, write $\delta_\mathcal{F}(d+1)$ for the proportion of $(d+1)$-element subfamilies having a common point, and let $\phi^{(d)}_\mathcal{F}(m)$ denote its $d$th homological shatter function.

Kalai–Meshulam's fractional Helly conjecture. There exists a function $\beta:(0,1)\to(0,1)$ such that, for every $\alpha>0$ and every sufficiently large set system $\mathcal{F}$ in $\mathbb{R}^d$ satisfying

$$\forall m\geq 0,\qquad \phi^{(d)}_\mathcal{F}(m)\leq A m^k,$$

if $\delta_\mathcal{F}(d+1)\geq\alpha$, then some $\beta(\alpha)|\mathcal{F}|$ members of $\mathcal{F}$ have a point in common.

This conjecture predicts a fractional Helly theorem for topological set systems whose homological shatter function has polynomial growth. The supplied passage presents it as one of two Kalai–Meshulam conjectures whose combination suggests this phenomenon; no resolution is given here.