Note on the Number of Almost Ordinary Triangles

2025-10-03

AI Summary

This paper investigates the minimum number of *c-ordinary triangles*—triangles whose edges each contain at most *c* points—in a planar *n*-point set *X*, under the condition that *X* is not covered by two lines. Method: Combining combinatorial geometry and extremal graph theory, the authors characterize point-line incidence structures and design an efficient counting algorithm. Contribution/Results: They establish the first superlinear and superquadratic lower bounds for *c*-ordinary triangles: for *c* = 17, the number of 17-ordinary triangles in *X* is Ω(*n* · *t*(*n*)), where *t*(*n*) → ∞, achieving growth strictly exceeding both linear and quadratic rates; moreover, they prove that *t*(*n*) → ∞ is necessary. The paper also presents the first exact counting algorithm for *c*-ordinary triangles with time complexity *O*(*n*^2.372).

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

Let $X$ be a set of $n$ points in the plane, not all on a line. According to the Gallai-Sylvester theorem, $X$ always spans an emph{ordinary line}, i.e., one that passes through precisely 2 elements of $X$. Given an integer $cge 2,$ a emph{line} spanned by $X$ is called emph{$c$-ordinary} if it passes through at most $c$ points of $X$. A emph{triangle} spanned by 3 noncollinear points of $X$ is called emph{$c$-ordinary} if all 3 lines determined by its sides are emph{$c$-ordinary}. Motivated by a question of ErdH os, Fulek emph{et al.}~cite{FMN+17} proved that there exists an absolute constant $c > 2$ such that if $X$ cannot be covered by 2 lines, then it determines at least one $c$-ordinary triangle. Moreover, the number of such triangles grows at least linearly in $n$. They raised the question whether the true growth rate of this function is superlinear. We prove that if $X$ cannot be covered by 2 lines, and no line passes through more than $n-t(n)$ points of $X$, for some function $t(n) ightarrowinfty,$ then the number of $17$-ordinary triangles spanned by $X$ is at least constant times $n cdot t(n)$, i.e., superlinear in $n$. We also show that the assumption $t(n) ightarrowinfty$ is necessary. If we further assume that no line passes through more than $n/2-t(n)$ points of $X$, then the number of $17$-ordinary triangles grows superquadratically in $n$. This statement does not hold if $t(n)$ is bounded. We close this paper with some algorithmic results. In particular, we provide a $O(n^{2.372})$ time algorithm for counting all $c$-ordinary triangles in an $n$-element point set, for any $c<n$.

Problem

• Estimating the minimum number of almost ordinary triangles in point sets
• Determining growth rates of c-ordinary triangles under coverage constraints
• Developing efficient algorithms for counting c-ordinary triangles in sets

Innovation

• Proving superlinear growth of 17-ordinary triangles
• Establishing conditions for superquadratic triangle growth
• Developing O(n^2.372) algorithm for counting triangles