Sharp supersaturation order for pentagons in linear triple systems

Establish whether every n-vertex linear triple system with m edges, in the range m much larger than n^{3/2}, contains Θ(m^6/n^7) copies of the loose 5-cycle C_5, thereby determining whether the lower bound in Theorem 1.1 is sharp in order of magnitude.

Background

Theorem 1.1 proves that an n-vertex linear triple system with m > 100n^{3/2} edges contains at least m^6/n^7 copies of the pentagon C_5. The paper explains that an initially proposed stronger bound, m^5/n^5, cannot hold in general, based on a construction communicated by Methuku.

The authors nevertheless conjecture that the proved lower bound has the correct order of magnitude, namely Θ(m^6/n^7). This remains unresolved and is restated later as Conjecture 2.1 for the range n^{3/2} ≪ m ≪ n^2.