Subset Selection Problems in Planar Point Sets
József Balogh, Felix Christian Clemen, Adrian Dumitrescu, Dingyuan Liu
TL;DR
This work studies three subset-selection problems in planar point sets under the constraint that at most $s$ points lie on any line: large general-position subsets, monotone general-position subsets, and subsets with pairwise distinct slopes. It develops a unified framework leveraging incidence geometry, probabilistic methods, the hypergraph container method, and additive combinatorics (e.g., Sidon sets) to obtain regime-dependent bounds across $n$ and $s$. Key contributions include new upper and lower bounds for $f(n,s)$, sharp-ish gaps for $g(n,s)$ in certain regimes, and polynomially nontrivial bounds for $h(n,s)$, along with consequences for planar Ramsey-type numbers and coloring schemes. The results deepen the understanding of how geometric constraints interact with combinatorial container methods and random constructions, with implications for no-three-in-line-type questions and related extremal geometry problems.
Abstract
Given a finite set satisfying condition $\mathcal{A}$, the subset selection problem asks, how large of a subset satisfying condition $\mathcal{B}$ can we find? We make progress on three instances of subset selection problems in planar point sets. Let $n,s\in\mathbb{N}$ with $n\geq s$, and let $P\subseteq\mathbb{R}^2$ be a set of $n$ points, where at most $s$ points lie on the same line. Firstly, we select a general position subset of $P$, i.e., a subset containing no $3$ points on the same line. This problem was proposed by Erdős under the regime when $s$ is a constant. For $s$ being non-constant, we give new lower and upper bounds on the maximum size of such a subset. In particular, we show that in the worst case such a set can have size at most $O(n/s)$ when $n^{1/3}\leq s\leq n$ and $O(n^{5/6+o(1)}/\sqrt{s})$ when $3\leq s\leq n^{1/3}$. Secondly, we select a monotone general position subset of $P$, that is, a subset in general position where the points are ordered from left to right and their $y$-coordinates are either non-decreasing or non-increasing. We present bounds on the maximum size of such a subset. In particular, when $s=Θ(\sqrt{n})$, our upper and lower bounds differ only by a logarithmic factor. Lastly, we select a subset of $P$ with pairwise distinct slopes. This problem was initially studied by Erdős, Graham, Ruzsa, and Taylor on the grid. We show that for $s=O(\sqrt{n})$ such a subset of size $Ω((n/\log{s})^{1/3})$ can always be found in $P$. When $s=Θ(\sqrt{n})$, this matches a lower bound given by Zhang on the grid. As for the upper bound, we show that in the worst case such a subset has size at most $O(\sqrt{n})$ for $2\leq s\leq n^{3/8}$ and $O((n/s)^{4/5})$ for $n^{3/8}\leq s=O(\sqrt{n})$. The proofs use a wide range of tools such as incidence geometry, probabilistic methods, the hypergraph container method, and additive combinatorics.