Prove the unresolved cases of the Erdős girth conjecture

Prove the Erdős girth conjecture asserting that, for every positive integer \(\kappa\), there exist \(n\)-vertex graphs of girth \(2\kappa+2\) with \(\Omega(n^{1+1/\kappa})\) edges, including all values of \(\kappa\) not covered by the currently known cases.

Background

The paper explains that dense high-girth graphs yield lower bounds for explicit multiplicative spanner construction because edges lying on no short cycle are mandatory in every valid spanner. The Erdős girth conjecture would provide asymptotically tight dense graphs of the required girth, but it is known only for κ∈{1,2,3,5}\kappa\in\{1,2,3,5\}; in particular, the case κ=4\kappa=4, corresponding to girth ten, remains unresolved.

References

The girth conjecture of Erd\H{o}s asserts that there are $n$-vertex graphs of girth $2\kappa+2$ with $\Omega(n{1+1/\kappa})$ edges, which would be tight . The conjecture is proven exactly for $\kappa\in{1,2,3,5}$ and remains unproven everywhere else including $\kappa=4$.

— Quantum Query Lower Bounds for Triangle-Listing and Spanners  (2609.37091 - Chen et al., 29 Sep 2026) in Section 1, paragraph “Multiplicative $k$-Spanner Construction”