AI 中文总结
研究在有禁用树的图中计算大团数量的问题，当$r = k - 2$或$r = k - 3\geq5$时，验证Gerbner和Palmer的猜想，证明${\rm ex}(n,K_r,T)=a\binom{k - 1}{r}+\binom{b}{r}$并刻画极值图。
AI 中文摘要
给定图$H$和$F$，广义图兰数${\rm ex}(n,H,F)$是$n$个顶点的不含$F$的图中$H$的最大副本数。Alon和Shikhelman发起了对广义图兰问题的系统研究。设$T$是有$k$个顶点的树，$n = a(k - 1) + b$，其中$0\leq b<k - 1$。Gerbner和Palmer猜想对于每个$r\geq3$，图$aK_{k - 1}\cup K_b$在所有$n$个顶点的不含$T$的图中使$K_r$的副本数最大化。本文验证了$r = k - 2$或$r = k - 3\geq5$时的猜想，表明${\rm ex}(n,K_r,T)=a\binom{k - 1}{r}+\binom{b}{r}$并刻画了所有极值图。
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. Let $T$ be a tree on $k$ vertices, and write $n=a(k-1)+b$, where $0\leq b<k-1$. Recently, Gerbner and Palmer (Electron. J. Combin., 2026) proposed the following conjecture: for every $r\geq3$, the graph $aK_{k-1}\cup K_b$ maximizes the number of copies of $K_r$ among all $n$-vertex $T$-free graphs. In this paper, we verify their conjecture when $r=k-2$ or $r=k-3\geq5$. More precisely, we show that ${\rm ex}(n,K_r,T)=a\binom{k-1}{r}+\binom{b}{r}$ and characterize all extremal graphs.
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