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Growth Rate of the Number of Empty Triangles in the Plane

Bhaswar B. Bhattacharya, Sandip Das, Sk Samim Islam, Saumya Sen

TL;DR

This work investigates how the number of empty triangles $N_{ riangle}(P)$ changes when a point $x$ is removed from a planar point set $P$ in general position. The authors translate the geometric problem into a graph-theoretic one by defining the empty-triangle incidence graph $G_P(x)$ and prove it is kite-free, allowing them to bound the change $\Delta(x,P)$ through triangle counts in $G_P(x)$. They derive an upper bound $\Delta(x,P)=o(V_P(x)^2)$ via connections to the Ruzsa–Szemerédi problem and provide a matching $\Omega(V_P(x)^{3/2})$ lower bound through a constructive configuration, highlighting a gap between these rates. They also show $G_P(x)$ can host large bipartite subgraphs but Behrend-type constructions are not geometrically realizable as $G_P(x)$, underscoring nontrivial geometric constraints. Overall, the paper bridges discrete geometry and extremal graph theory, proposing structural avenues to sharpen the bounds on $\Delta(x,P)$ in future work.

Abstract

Given a set $P$ of $n$ points in the plane, in general position, denote by $N_Δ(P)$ the number of empty triangles with vertices in $P$. In this paper we investigate by how much $N_Δ(P)$ changes if a point $x$ is removed from $P$. By constructing a graph $G_P(x)$ based on the arrangement of the empty triangles incident on $x$, we transform this geometric problem to the problem of counting triangles in the graph $G_P(x)$. We study properties of the graph $G_P(x)$ and, in particular, show that it is kite-free. This relates the growth rate of the number of empty triangles to the famous Ruzsa-Szemerédi problem.

Growth Rate of the Number of Empty Triangles in the Plane

TL;DR

This work investigates how the number of empty triangles changes when a point is removed from a planar point set in general position. The authors translate the geometric problem into a graph-theoretic one by defining the empty-triangle incidence graph and prove it is kite-free, allowing them to bound the change through triangle counts in . They derive an upper bound via connections to the Ruzsa–Szemerédi problem and provide a matching lower bound through a constructive configuration, highlighting a gap between these rates. They also show can host large bipartite subgraphs but Behrend-type constructions are not geometrically realizable as , underscoring nontrivial geometric constraints. Overall, the paper bridges discrete geometry and extremal graph theory, proposing structural avenues to sharpen the bounds on in future work.

Abstract

Given a set of points in the plane, in general position, denote by the number of empty triangles with vertices in . In this paper we investigate by how much changes if a point is removed from . By constructing a graph based on the arrangement of the empty triangles incident on , we transform this geometric problem to the problem of counting triangles in the graph . We study properties of the graph and, in particular, show that it is kite-free. This relates the growth rate of the number of empty triangles to the famous Ruzsa-Szemerédi problem.
Paper Structure (7 sections, 7 theorems, 9 equations, 9 figures)

This paper contains 7 sections, 7 theorems, 9 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['triangle_containig_x']}
  3. Proof of the Upper Bound
  4. Lower Bound Construction
  5. Properties of the Graph $G_P(x)$
  6. Proof of Proposition \ref{['ppn:geometric']}
  7. Conclusions

Key Result

theorem thmcountertheorem

For any set $P$, with $|P| =n$, where $H(V_P(x), K_3, K_4\setminus\{e\})$ is the maximum number of triangles in a $K_4\setminus\{e\}$-free graph on $V_P(x)$ vertices. Moreover, there exists a set $P$, with $|P|=n$, and a point $x \in P$ such that $\Delta(x, P) \geq C V_P(x)^{\frac{3}{2}}$, for some constant $C > 0$.

Figures (9)

  • Figure 1: The kite graph $K_4 \setminus \{ e \}$.
  • Figure 2: A set of points $P=\{x, a, b, c\}$ and the empty triangle graph incident at $x$.
  • Figure 3: Illustration for the proof of Lemma \ref{['counting_N_22']}.
  • Figure 4: Example showing the lower bound in Theorem \ref{['triangle_containig_x']}.
  • Figure 5: Illustration for the proof of Lemma \ref{['lm:bipartite']}.
  • ...and 4 more figures

Theorems & Definitions (14)

  • theorem thmcountertheorem
  • proof
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • definition thmcounterdefinition: Behrend's graph
  • proposition thmcounterproposition
  • ...and 4 more