  Two sets $A$ and $B$ of points in the plane are \emph{mutually avoiding} if
no line generated by any two points in $A$ intersects the convex hull of $B$,
and vice versa. In 1994, Aronov, Erd\H os, Goddard, Kleitman, Klugerman, Pach,
and Schulman showed that every set of $n$ points in the plane in general
position contains a pair of mutually avoiding sets each of size at least
$\sqrt{n/12}$. As a corollary, their result implies that for every set of $n$
points in the plane in general position one can find at least $\sqrt{n/12}$
segments, each joining two of the points, such that these segments are pairwise
crossing. In this note, we prove a fractional version of their theorem: for
every $k > 0$ there is a constant $\varepsilon_k > 0$ such that any
sufficiently large point set $P$ in the plane contains $2k$ subsets
$A_1,\ldots, A_{k},B_1,\ldots, B_k$, each of size at least $\varepsilon_k|P|$,
such that every pair of sets $A = \{a_1,\ldots, a_k\}$ and $B = \{b_1,\ldots,
b_k\}$, with $a_i \in A_i$ and $b_i \in B_i$, are mutually avoiding. Moreover,
we show that $\varepsilon_k = \Omega(1/k^4)$. Similar results are obtained in
higher dimensions
