  Two subsets $A,B$ of an $n$-element ground set $X$ are said to be
\emph{crossing}, if none of the four sets $A\cap B$, $A\setminus B$,
$B\setminus A$ and $X\setminus(A\cup B)$ are empty. It was conjectured by
Karzanov and Lomonosov forty years ago that if a family $\mathcal{F}$ of
subsets of $X$ does not contain $k$ pairwise crossing elements, then
$|\mathcal{F}|=O_{k}(n)$. For $k=2$ and $3$, the conjecture is true, but for
larger values of $k$ the best known upper bound, due to Lomonosov, is
$|\mathcal{F}|=O_{k}(n\log n)$. In this paper, we improve this bound by showing
that $|\mathcal{F}|=O_{k}(n\log^{*} n)$ holds, where $\log^{*}$ denotes the
iterated logarithm function.
