2025/01/27 by Mubayi, Dhruv, Solymosi, Jozsef
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2501.15861
We prove that every n vertex linear triple system with m edges has at least m6/n7 copies of a pentagon, provided m>100 n3/2. This provides the first nontrivial bound for a question posed by Jiang and Yepremyan. More generally, for each ℓ ≥ 2, we prove that there is a constant c such that if an n-vertex graph is ε-far from being triangle-free, with ε ≫ n-1/3ℓ, then it has at least c ε3ℓ n2ℓ+1 copies of C2ℓ+1. This improves the previous best bound of c ε4ℓ+2 n2ℓ+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 n11/6 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≈ 1.83 cannot be reduced to anything smaller than log3 6 ≈ 1.726.