For a finite set $A\subset \mathbb{R}^d$, let $Δ(A)$ denote the spread of $A$, which is the ratio of the maximum pairwise distance to the minimum pairwise distance. For a positive integer $n$, let $γ_d(n)$ denote the largest integer such that any set $A$ of $n$ points in general position in $\mathbb{R}^d$, satisfying $Δ(A) \leq αn^{1/d}$ for a fixed $α>0$, contains at least $γ_d(n)$ points in convex position. About $30$ years ago, Valtr proved that $γ_2(n)=Θ(n^{1/3})$. Since then no further results have been obtained in higher dimensions. Here we continue this line of research in three dimensions and prove that $γ_3(n) =Θ(n^{1/2})$. The lower bound implies the following approximation: Given any $n$-element point set $A\subset \mathbb{R}^3$ in general position, satisfying $Δ(A) \leq αn^{1/3}$ for a fixed $α$, a $Ω(n^{-1/6})$-factor approximation of the maximum-size convex subset of points can be computed by a randomized algorithm in $O(n \log{n})$ expected time.
Let $\mathcal{T}$ be an unrooted binary tree with $n$ distinctly labelled leaves. Deriving its name from the field of phylogenetics, a convex character on $\mathcal{T}$ is simply a partition of the leaves such that the minimal spanning subtrees induced by the blocks of the partition are mutually disjoint. In earlier work Kelk and Stamoulis (Advances in Applied Mathematics 84 (2017), pp. 34--46) defined $g_k(\mathcal{T})$ as the number of convex characters where each block has at least $k$ leaves. Exact expressions were given for $g_1$ and $g_2$, where the topology of $\mathcal{T}$ turns out to…
According to Suk's breakthrough result on the Erdos-Szekeres problem, any point set in general position in the plane, which has no $n$ elements that form the vertex set of a convex $n$-gon, has at most $2^{n+O\left({n^{2/3}\log n}\right)}$ points. We strengthen this theorem in two ways. First, we show that the result generalizes to convexity structures induced by pseudoline arrangements. Second, we improve the error term. A family of $n$ convex bodies in the plane is said to be in convex position if the convex hull of the union of no $n-1$ of its members contains the remaining one. If any thre…
We call $i$ a fixed point of a given sequence if the value of that sequence at the $i$-th position coincides with $i$. Here, we enumerate fixed points in the class of restricted growth sequences. The counting process is conducted by calculation of generating functions and leveraging a probabilistic sampling method.
Let $ES_{d}(n)$ be the smallest integer such that any set of $ES_{d}(n)$ points in $\mathbb{R}^{d}$ in general position contains $n$ points in convex position. In 1960, Erdős and Szekeres showed that $ES_{2}(n) \geq 2^{n-2} + 1$ holds, and famously conjectured that their construction is optimal. This was nearly settled by Suk in 2017, who showed that $ES_{2}(n) \leq 2^{n+o(n)}$. In this paper, we prove that $$ES_{d}(n) = 2^{o(n)}$$ holds for all $d \geq 3$. In particular, this establishes that, in higher dimensions, substantially fewer points are needed in order to ensure the presence of a con…
