For all integers l>k≥ 2, let mathscrC_2k=C_3,C_4,…,C_2k. Then operatornameexbigl(n,C_2k+1,mathscrC_2kcupC_2l+1bigr)=o(n^2)qquadas n→∞, where operatornameex(n,H,F) is the maximum number of copies of H in an n-vertex graph containing no member of F, and C_i denotes the cycle of length i.

Gerbner–Győri–Methuku–Vizer generalized Turán conjecture

For all integers $l>k\ge 2$ , let $\mathscr{C}_{2k}=\{C_3,C_4,\ldots,C_{2k}\}$ . Then

\operatorname{ex}\bigl(n,C_{2k+1},\mathscr{C}_{2k}\cup\{C_{2l+1}\}\bigr)=o(n^2)\qquad\text{as }n\to\infty,

where $\operatorname{ex}(n,H,\mathcal{F})$ is the maximum number of copies of $H$ in an $n$ -vertex graph containing no member of $\mathcal{F}$ , and $C_i$ denotes the cycle of length $i$ .

References

Primary source

arXiv

Additional references

A subquadratic bound for generalized Turán numbers of odd cycles — arXiv

combinatorics extremal-graph-theory

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A new unrefereed preprint claims to prove the conjecture, but the result has not yet been independently verified.

The conjecture, posed by Gerbner, Győri, Methuku, and Vizer in 2017, predicts a subquadratic bound for counting one odd cycle while forbidding an even cycle and a longer odd cycle.

Known results

Gerbner, Győri, Methuku, and Vizer (2017) proved only the quadratic upper bound $O(n^2)$ for $k<l$ , with lower bound $\Omega(n^{1+1/(2l+2)})$ .

August 24, 2026 claimed proof

A preprint, A subquadratic bound for generalized Turán numbers of odd cycles, claims that for all $l>k\ge 2$ the conjectured subquadratic generalized Turán bound holds. This would settle the conjecture, but the preprint is unrefereed and the claim remains unverified.

Current status (as of August 2026): the conjecture has a claimed proof in an unrefereed preprint, but independent verification is absent.

Sources

No solutions have been posted yet.