Many pentagons in triple systems

Abstract: We prove that every $n$ vertex linear triple system with $m$ edges has at least $m<sup>6/n<sup>7$ copies of a pentagon, provided $m&gt;100 \, n<sup>{3/2}$. 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 $\varepsilon \gg n<sup>{-1/3\ell}$, then it has at least $c \, \varepsilon^{3\ell} n<sup>{2\ell+1}$ copies of $C_{2\ell+1}$. This improves the previous best bound of $$c \, \varepsilon^{4\ell+2} n<sup>{2\ell+1}$$ 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 $60\, n<sup>{11/6}$ 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 $\log_3 6 \approx 1.726$.