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arXiv · 1601.07206

Sets in Almost General Position

Luka Milićević
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Abstract

Erdős asked the following question: given $n$ points in the plane in almost general position (no 4 collinear), how large a set can we guarantee to find that is in general position (no 3 collinear)? Füredi constructed a set of $n$ points in almost general position with no more than $o(n)$ points in general position. Cardinal, Tóth and Wood extended this result to $\mathbb{R}^3$, finding sets of $n$ points with no 5 on a plane whose subsets with no 4 points on a plane have size $o(n)$, and asked the question for higher dimensions: for given $n$, is it still true that the largest subset in general position we can guarantee to find has size $o(n)$? We answer their question for all $d$ and derive improved bounds for certain dimensions.

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Luka Milićević. 2016-01-26. Sets in Almost General Position. https://arxiv.org/abs/1601.07206

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