Sharpness of the Bondy–Simonovits bound

Determine, for integers k outside {2,3,5}, whether there exist C_{2k}-free graphs on n vertices with Ω(n^{1+1/k}) edges, thereby establishing the sharpness of the Bondy–Simonovits exponent.

Background

The Bondy–Simonovits Theorem gives ex(n,C_{2k})=O(n{1+1/k}) for every fixed k≥2. The paper notes that matching lower bounds are known only for k∈{2,3,5}; determining whether the same order of magnitude can be attained for other even cycle lengths is a central extremal graph-theoretic problem.

References

At the moment, this bound is only known to be tight when $k \in \{2,3,5\}$ . For other values of~$k$, it is a major open problem to determine the existence of $C_{2k}$-free graphs with $\Omega(n{1+1/k})$ edges, where $C_{2k}$ denotes the cycle on $2k$ vertices.

— Generic solutions to symmetric linear equations  (2610.02177 - Frederickson et al., 1 Oct 2026) in Section 1, subsection “Our methods: revisiting Bondy–Simonovits”