Many pentagons in triple systems

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 and improve this further to for a 3‐partite version of the problem.