Lower bounds for odd-cycle Turán numbers of linear triple systems

Determine corresponding lower bounds for ex_L(n,C_{2ℓ+1}) when ℓ > 2, where ex_L(n,C_{2ℓ+1}) denotes the maximum number of edges in an n-vertex linear triple system containing no copy of the loose odd cycle C_{2ℓ+1}.

Background

The paper states that the upper bound ex_L(n,C_{2ℓ+1}) = O(n{1+1/ℓ}) is known for all relevant ℓ, while a matching or otherwise corresponding lower bound is not known for any ℓ > 2. Thus, the unresolved issue concerns the construction of dense C_{2ℓ+1}-free linear triple systems for longer odd cycles.

References

More generally, the upper bound exL(n, C2ℓ+1) = O(n1+1/ℓ) was proved in [4]; no corresponding lower bound is known for any ℓ > 2.

— Many pentagons in triple systems  (2501.15861 - Mubayi et al., 27 Jan 2025) in Introduction, paragraph discussing extremal numbers (PDF p. 2)