An Improved Upper Bound for the Turán Number of the Hexagon

RESUMO

For a graph $F$, the Turán number $\operatorname{ex}(n,F)$ is the maximum number of edges in an $n$-vertex graph containing no copy of $F$. Determining the Turán numbers of even cycles is a central problem in extremal graph theory and remains open in general. For $C_6$, the best previous upper bound was due to Füredi, Naor, and Verstraëte [Advances in Mathematics, 2006], who proved that, for sufficiently large positive integer $n$, $$ \operatorname{ex}(n,C_6) \leq \lambda n^{4/3}+O(n)<0.6272 n^{4/3}, $$ where $\lambda$ is the real root of $ 16\lambda^3-4\lambda^2+\lambda-3=0$. We improve this bound by showing that, for sufficiently large positive integer $n$, $$ \operatorname{ex}(n,C_6) \leq \alpha n^{4/3}+O(n)<0.6144 n^{4/3}, $$ where $\alpha$ is the unique real root of $ 4 \alpha^{3} (3/2)^{1-1/(2\alpha)} =1$ in the interval $(1/2,2/3)$.