AI 中文总结
本研究针对广义Turán问题中关于色数与树宽的猜想构造反例，证明对所有r≥3存在满足色数和树宽均为r的图F_r，使得ex(n,K_r,F_r)为o(n^{r-1})，证否了相关猜想与综述中的对应结论。
AI 中文摘要
给定图$H$和$F$，广义Turán数${\rm ex}(n,H,F)$是$n$个顶点的无$F$图中$H$副本的最大数量。Alon和Shikhelman（J. Combin. Theory Ser. B, 2016）开启了广义Turán问题的系统研究。近期，Gao、Wu和Xue（J. Graph Theory, 2026）提出疑问：每个色数$χ(F)=r\geq3$且树宽${\rm tw}(F)\geq r$的图$F$是否都满足${\rm ex}(n,K_r,F)=Ω(n^{r-1})$。在本注记中，我们对所有$r\geq3$的情况给出了该问题的否定答案。更确切地说，我们证明图$F_r=K_{r-3}\vee H$（其中$H$由$K_4$的一条边细分一次得到）满足$χ(F_r)={\rm tw}(F_r)=r$，且有\\[ n^{r-1}e^{-O(\sqrt{\log n})}\leq {\rm ex}(n,K_r,F_r)=o(n^{r-1}). \\]该结果也证否了Gerbner和Palmer近期综述（Electron. J. Combin., 2026）中的猜想6.3。
英文摘要
Given graphs $H$ and $F$, the generalized Turán number ${\rm ex}(n,H,F)$ is the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. Alon and Shikhelman (J. Combin. Theory Ser. B, 2016) initiated the systematic study of generalized Turán problems. Recently, Gao, Wu and Xue (J. Graph Theory, 2026) asked whether every graph $F$ with chromatic number $χ(F)=r\geq3$ and treewidth ${\rm tw}(F)\geq r$ satisfies ${\rm ex}(n,K_r,F)=Ω(n^{r-1})$. In this note, we give a negative answer to this question for every $r\geq3$. More precisely, we prove that the graph $F_r=K_{r-3}\vee H$, where $H$ is obtained from $K_4$ by subdividing one edge once, satisfies $χ(F_r)={\rm tw}(F_r)=r$ and \[ n^{r-1}e^{-O(\sqrt{\log n})}\leq {\rm ex}(n,K_r,F_r)=o(n^{r-1}). \] This result also disproves Conjecture 6.3 in the recent survey of Gerbner and Palmer (Electron. J. Combin., 2026).
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