Many pentagons in triple systems

TL;DR

This paper establishes new lower bounds on the number of pentagons in triple systems and odd cycles in graphs, with applications to geometric configurations, advancing understanding in combinatorics and geometry.

Contribution

It provides the first nontrivial bound for pentagon copies in triple systems and improves bounds on odd cycle counts in graphs, also connecting to geometric theorems.

Findings

Lower bound of m^6/n^7 for pentagons in triple systems

Improved bound for odd cycles in graphs from ε^{4ℓ+2} to ε^{3ℓ}

Geometric result on triangles sharing a harmonic point in large point sets

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 $\log_3 6 \approx 1.726$.