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- W4301260891 abstract "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, ErdH 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" @default.
- W4301260891 created "2022-10-05" @default.
- W4301260891 creator A5068093083 @default.
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- W4301260891 date "2018-02-18" @default.
- W4301260891 modified "2023-10-14" @default.
- W4301260891 title "A positive fraction mutually avoiding sets theorem" @default.
- W4301260891 doi "https://doi.org/10.48550/arxiv.1802.06484" @default.
- W4301260891 hasPublicationYear "2018" @default.
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