Many pentagons in triple systems

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 $\log_3 6 \approx 1.726$.