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Many pentagons in triple systems

Dhruv Mubayi, Jozsef Solymosi

TL;DR

This work resolves a supersaturation question for odd cycles in linear 3-graphs by proving that dense linear triple systems contain many pentagons: for $n>10$ and $m>100\,n^{3/2}$, a linear $n$-vertex triple system with $m$ edges has at least $m^6/n^7$ copies of $C_5$. The authors develop a general framework using shadow graphs to extend to longer odd cycles, obtaining a bound of $\Omega\big(m^{3k}/n^{4k-1}\big)$ copies of $C_{2k+1}$ for each $k\ge2$, with $\,m\gg n^{2-1/(3k)}$. They connect these combinatorial results to geometry by showing that sets with many triangles similar to a fixed triangle yield harmonic-point configurations, and they demonstrate near-optimal exponent behavior via explicit constructions (e.g., a pentagon-free, linear 3-graph with many edges) and a Ruzsa-type geometric construction giving many triangles with disjoint harmonic points. Finally, they derive a density-removal-type consequence: if a graph is $\varepsilon$-far from triangle-free, then it contains at least $c\,\varepsilon^{3\ell} n^{2\ell+1}$ copies of $C_{2\ell+1}$ for every fixed $\ell\ge2$, linking extremal and geometric themes for odd cycles in graphs and 3-graphs.

Abstract

We prove that every $n$ vertex linear triple system with $m$ edges has at least $m^6/n^7$ copies of a pentagon, provided $m>100 \, n^{3/2}$. This provides the first nontrivial bound for a question posed by Jiang and Yepremyan. More generally, for each $ \ell \ge 2$, we prove that there is a constant $c$ such that if an $n$-vertex graph is $\varepsilon$-far from being triangle-free, with $\varepsilon \gg n^{-1/3\ell}$, then it has at least $c \, \varepsilon^{3\ell} n^{2\ell+1}$ copies of $C_{2\ell+1}$. This improves the previous best bound of $c \, \varepsilon^{4\ell+2} n^{2\ell+1}$ due to Gishboliner, Shapira and Wigderson. Our result also yields some geometric theorems, including the following. For $n$ large, every $n$-point set in the plane with at least $60\, n^{11/6}$ triangles similar to a given triangle $T$, contains two triangles sharing a special point, called the harmonic point. In the other direction, we give a construction showing that the exponent $11/6\approx 1.83$ cannot be reduced to anything smaller than $\log_3 6 \approx 1.726$.

Many pentagons in triple systems

TL;DR

This work resolves a supersaturation question for odd cycles in linear 3-graphs by proving that dense linear triple systems contain many pentagons: for and , a linear -vertex triple system with edges has at least copies of . The authors develop a general framework using shadow graphs to extend to longer odd cycles, obtaining a bound of copies of for each , with . They connect these combinatorial results to geometry by showing that sets with many triangles similar to a fixed triangle yield harmonic-point configurations, and they demonstrate near-optimal exponent behavior via explicit constructions (e.g., a pentagon-free, linear 3-graph with many edges) and a Ruzsa-type geometric construction giving many triangles with disjoint harmonic points. Finally, they derive a density-removal-type consequence: if a graph is -far from triangle-free, then it contains at least copies of for every fixed , linking extremal and geometric themes for odd cycles in graphs and 3-graphs.

Abstract

We prove that every vertex linear triple system with edges has at least copies of a pentagon, provided . This provides the first nontrivial bound for a question posed by Jiang and Yepremyan. More generally, for each , we prove that there is a constant such that if an -vertex graph is -far from being triangle-free, with , then it has at least copies of . This improves the previous best bound of due to Gishboliner, Shapira and Wigderson. Our result also yields some geometric theorems, including the following. For large, every -point set in the plane with at least triangles similar to a given triangle , contains two triangles sharing a special point, called the harmonic point. In the other direction, we give a construction showing that the exponent cannot be reduced to anything smaller than .
Paper Structure (8 sections, 8 theorems, 58 equations, 6 figures)

This paper contains 8 sections, 8 theorems, 58 equations, 6 figures.

Table of Contents

  1. Introduction
  2. An application in geometry
  3. Proof of Theorem \ref{['thm:mainC5']}
  4. Constructions of pentagon-free triple systems
  5. Generalization to longer cycles
  6. Proof of Theorem \ref{['thm:geom']}
  7. A geometric construction
  8. Proof of Theorem \ref{['thm:densetriangles']}

Key Result

Theorem 1.1

Let $n>10$ and let $H$ be an $n$-vertex linear triple system with $m > 100 \, n^{3/2}$ edges. Then the number of copies of $C_5$ in $H$ is at least $m^6/n^7$.

Figures (6)

  • Figure 1: $A,C,E$ are the harmonic points of the equilateral triangle $BDF$. $H, J, L$ are the harmonic points of the isosceles right triangle $GIK$.
  • Figure 2: A good path $wxyz$ in $G_u$ and expansion of $uaxyc$
  • Figure 3: The multigraph $G_u$
  • Figure 4: A 7-pseudocycle $C_v$
  • Figure 5: Triangles forming a pentagon
  • ...and 1 more figures

Theorems & Definitions (12)

  • Theorem 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • proof
  • Conjecture 2.1
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • ...and 2 more