发表机构
Indian Statistical Institute（印度统计研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究改进了六边形（C6）的Turán数上界，通过引入新常数α，将上界从0.6272n^{4/3}降至0.6144n^{4/3}，推进了偶圈极值图论的研究。
AI 中文摘要
对于图$F$，Turán数$\operatorname{ex}(n,F)$是不含$F$副本的$n$顶点图的最大边数。确定偶圈的Turán数是极值图论中的一个核心问题，并且总体上仍未解决。对于$C_6$，之前最好的上界归功于Füredi、Naor和Verstraëte [Advances in Mathematics, 2006]，他们证明了对于足够大的正整数$n$，$$ \operatorname{ex}(n,C_6) \leq \lambda n^{4/3}+O(n)<0.6272 n^{4/3}, $$ 其中$\lambda$是方程$ 16\lambda^3-4\lambda^2+\lambda-3=0$的实根。我们通过证明对于足够大的正整数$n$，$$ \operatorname{ex}(n,C_6) \leq \alpha n^{4/3}+O(n)<0.6144 n^{4/3}, $$ 来改进这一界，其中$\alpha$是方程$ 4 \alpha^{3} (3/2)^{1-1/(2\alpha)} =1$在区间$(1/2,2/3)$内的唯一实根。
英文摘要
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 $n$, $$ \operatorname{ex}(n,C_6) \leq λn^{4/3}+O(n)<0.6272 n^{4/3}, $$ where $λ$ is the real root of $ 16λ^3-4λ^2+λ-3=0$. We improve this bound by showing that, for sufficiently large positive integer $n$, $$ \operatorname{ex}(n,C_6) \leq αn^{4/3}+O(n)<0.6144 n^{4/3}, $$ where $α$ is the unique real root of $ 4 α^{3} (3/2)^{1-1/(2α)} =1$ in the interval $(1/2,2/3)$.
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