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On higher dimensional point sets in general position
- Part of
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- Journal:
- Combinatorics, Probability and Computing , First View
- Published online by Cambridge University Press:
- 03 November 2025, pp. 1-15
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- Article
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A finite point set in
$\mathbb{R}^d$ is in general position if no 
$d + 1$ points lie on a common hyperplane. Let 
$\alpha _d(N)$ be the largest integer such that any set of 
$N$ points in 
$\mathbb{R}^d$, with no 
$d + 2$ members on a common hyperplane, contains a subset of size 
$\alpha _d(N)$ in general position. Using the method of hypergraph containers, Balogh and Solymosi showed that 
$\alpha _2(N) \lt N^{5/6 + o(1)}$. In this paper, we also use the container method to obtain new upper bounds for 
$\alpha _d(N)$ when 
$d \geq 3$. More precisely, we show that if 
$d$ is odd, then 
$\alpha _d(N) \lt N^{\frac {1}{2} + \frac {1}{2d} + o(1)}$, and if 
$d$ is even, we have 
$\alpha _d(N) \lt N^{\frac {1}{2} + \frac {1}{d-1} + o(1)}$. We also study the classical problem of determining 
$a(d,k,n)$, the maximum number of points selected from the grid 
$[n]^d$ such that no 
$k + 2$ members lie on a 
$k$-flat, and improve the previously best known bound for 
$a(d,k,n)$, due to Lefmann in 2008, by a polynomial factor when 
$k$ = 2 or 3 (mod 4).
