Constant-factor determination of \(\pi(C_5(r))\)

Determine \(\pi(C_5(r))\) within a constant factor as \(r\to\infty\), where \(C_5(r)\) is the \(r\)-augmentation of the 5-cycle.

Background

For every fixed 3-chromatic graph, the paper proves a lower bound of order 1/[r(log⁡r)1+o(1)]1/[r(\log r)^{1+o(1)}] for the Turán density of its rr-augmentation, while the general upper bound is $1/r$. For the 5-cycle C5C_5, these estimates differ by a multiplicative factor of (log⁡r)1+o(1)(\log r)^{1+o(1)}. The authors explicitly leave closing this gap as an open problem.

References

In the next simplest case when $F$ is the $5$-cycle $C_5$, we have multiplicative gap of $(\log r){1+o(1)}$ between the bounds and closing it is an interesting open problem.

— Turán densities of hypergraph augmentations  (2610.11757 - Pikhurko et al., 8 Oct 2026) in Problem 1, Section 5, Concluding remarks