Erdős–Sós问题中的大团与团谱半径
Large Cliques and Clique Spectral Radius in the Erdős--Sós Problem
AI总结：
本文针对Erdős–Sós问题，证明Gerbner和Palmer关于树T_t-自由图中r-团最大数量的猜想，还推导团谱半径的相关不等式，覆盖叶簇与少叶两种树类型。
AI中文摘要：
对于图H和F，令ex(n,H,F)为n顶点F-自由图中H的最大副本数。我们研究当H为团、F为t顶点固定树T_t时的该问题。Erdős–Sós猜想涉及ex(n,K_2,T_t)的值，Gerbner和Palmer提出更一般的猜想：若n=α(t-1)+β且0≤β≤t-2，则图αK_{t-1}⊔K_β在所有n顶点T_t-自由图中，对每个3≤r≤t-2都使r-团的数量最大。我们证明该猜想对具有至少t-r个共享父节点的叶节点的T_t成立，其中星型情况为特例，且可得到Gan、Loh和Sudakov猜想的精确团计数结果，该结果已由Chase及Chao和Dong证明。我们还研究团谱类似问题：在相同叶簇条件下，当n≥t-1时，每个T_t-自由图G满足ρ_r(G)≤C(t-2,r-1)，等号成立当且仅当K_{t-1}是G的一个分量。此外，当d≥2且t≥d²-d+3时，我们证明r=t-d时的猜想；对每个t，d=1的情况已完全确定。当d≥2且t≥d²-d+3时，每个T_t-自由图G满足ρ_{t-d}(G)≤ρ_{t-d}(K_{t-1})，等号成立当且仅当存在K_{t-1}分量。我们的方法针对相对较大的团：在少叶情况下，删除不在任何(t-d)-团中的边后，研究(t-d)-团间的相交关系，证明其等价类诱导非平凡团支撑分量，且每个非平凡分量最多有t-1个顶点；在互补多叶情况下，叶簇准则将团计数问题简化为精确最大度受限团定理。
英文摘要：
For graphs $H$ and $F$, let $ex(n,H,F)$ be the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. We study this problem when $H$ is a clique and $F=T_t$which is a fixed tree on $t$ vertices. The Erdős--Sós conjecture concerns the value of $ex(n,K_2, T_t)$. Gerbner and Palmer proposed a more general conjecture: if $n=α(t-1)+β$ and $0\leβ\le t-2$, then the graph $αK_{t-1}\sqcup K_β$ maximizes the number of $r$-cliques among all $n$-vertex $T_t$-free graphs for every $3\le r\le t-2$. We show that this conjecture holds for $T_t$ having at least $t-r$ leaves with a common parent, which contains the star case as a special case and recovers the sharp clique-counting result conjectured by Gan, Loh and Sudakov and proved by Chase and Chao and Dong. We also study the clique-spectral analogue. Under the same leaf-bunch condition, every $T_t$-free graph $G$ satisfies $ρ_r(G)\le\binom{t-2}{r-1}$, with equality, for $n\ge t-1$, if and only if $K_{t-1}$ is a component of $G$. Furthermore, we prove the conjecture for $r=t-d$ whenever $d\ge2$ and $t\ge d^2-d+3$, while the case $d=1$ is determined exactly for every $t$. For $d\ge2$ and $t\ge d^2-d+3$, every $T_t$-free graph $G$ satisfies $ρ_{t-d}(G)\leρ_{t-d}(K_{t-1})$, with equality characterized by the presence of a $K_{t-1}$-component. Our method is designed for relatively large cliques. In the leaf-poor case, after deleting edges that lie in no $(t-d)$-clique, we study the intersection relation among $(t-d)$-cliques and show that its equivalence classes induce the nontrivial clique-supported components; furthermore, we show each non-trivial component has at most $t-1$\) vertices. In the complementary leaf-rich case, a leaf-bunch criterion reduces the clique-counting problem to the sharp bounded-maximum-degree clique theorem.
