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Sets in Almost General Position

Part of: Discrete geometry

Published online by Cambridge University Press:  18 April 2017

LUKA MILIĆEVIĆ
LUKA MILIĆEVIĆ*
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, Wilberforce Road, Cambridge CB3 0WB, UK (e-mail: lm497@cam.ac.uk)
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Abstract

Erdős asked the following question: given n points in the plane in almost general position (no four collinear), how large a set can we guarantee to find that is in general position (no three 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 ℝ3, finding sets of n points with no five in a plane whose subsets with no four points in 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.

MSC classification

Primary: 52C35: Arrangements of points, flats, hyperplanes
Secondary: 52C10: Erdős problems and related topics of discrete geometry
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 26 , Issue 5 , September 2017 , pp. 720 - 745
Copyright
Copyright © Cambridge University Press 2017 

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References

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