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- W2989615071 abstract "Two sets A and B of points in the plane are mutually avoiding if no line generated by any two points in A intersects the convex hull of B, and vice versa. In 1994, Aronov, Erdős, 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 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 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 εk>0 such that any sufficiently large point set P in the plane contains 2k subsets A1,…,Ak,B1,…,Bk, each of size at least εk|P|, such that every pair of sets A={a1,…,ak} and B={b1,…,bk}, with ai∈Ai and bi∈Bi, are mutually avoiding. Moreover, we show that εk=Ω(1∕k4). Similar results are obtained in higher dimensions." @default.
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- W2989615071 date "2020-03-01" @default.
- W2989615071 modified "2023-09-29" @default.
- W2989615071 title "A positive fraction mutually avoiding sets theorem" @default.
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- W2989615071 doi "https://doi.org/10.1016/j.disc.2019.111730" @default.
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