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A positive fraction mutually avoiding sets theorem

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 $\varepsilon_k = Ω(1/k^4)$. Similar results are obtained in higher dimensions