发表机构
School of Mathematics and Statistics, Gansu Center for Applied Mathematics, Lanzhou University; Yau Mathematical Sciences Center (YMSC), Tsinghua University（兰州大学数学与统计学院，甘肃省应用数学中心; 清华大学丘成桐数学科学中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文确定了谱超饱和问题中团和奇圈的最优阈值常数，分别证明团的最优阈值为$(1-1/r)\sqrt2$，奇圈的最优阈值为$1/\sqrt2$，并给出拷贝数的精确下界。
AI 中文摘要
设 $Y_{n,r,q}$ 为从 Turán 图 $T_{n,r}$ 的最大部内添加 $q$ 条两两不相交的边所得到的图，并设 $c(n,F)$ 为向 $T_{n,r}$ 添加一条边所创造的 $F$ 的最小拷贝数。Fang、Li、Lin 和 Ma 证明了：对每个满足 $\chi(F)=r+1$ 的色临界图 $F$，存在常数 $\delta_F>0$，使得对所有充分大的 $n$ 和所有 $1\le q\le \delta_F\sqrt{n}$，条件 $\lambda(G)\ge\lambda(Y_{n,r,q})$ 迫使 $G$ 包含至少 $q\\,c(n,F)$ 个 $F$ 的拷贝。界 $q=O(\sqrt{n})$ 在常数因子意义下是紧的，这与 Mubayi、Pikhurko 和 Yilma 的边设置中的线性阶 $n$ 形成对比，但对任何 $F$，精确常数 $\delta_F$ 此前未知。在本文中，基于 Fang、Li、Lin 和 Ma 的结构性结果，我们确定了当 $F$ 为团和奇圈时的阈值 $\delta_F$。对每个 $r\ge2$，记 $\delta_r:=(1-\tfrac1r)\sqrt2$，并证明：对每个 $\varepsilon>0$，若 $n$ 充分大且 $1\le q\le(\delta_r-\varepsilon)\sqrt n$，则每个 $n$ 顶点且满足 $\lambda(G)\ge\lambda(Y_{n,r,q})$ 的图 $G$ 包含至少 $q\\,c(n,K_{r+1})$ 个 $K_{r+1}$ 的拷贝，且 $\delta_r$ 是最优的。对于奇圈，阈值为 $1/\sqrt2$。对每个 $k\ge1$ 和 $\varepsilon>0$，若 $n$ 充分大且 $1\le q\le(1/\sqrt2-\varepsilon)\sqrt n$，则每个 $n$ 顶点且满足 $\lambda(G)\ge\lambda(Y_{n,2,q})$ 的图 $G$ 包含至少 $q\\,c(n,C_{2k+1})$ 个 $C_{2k+1}$ 的拷贝，且 $1/\sqrt2$ 是最优的。我们的结果确定了拷贝的精确数量和 $q$ 的最优范围。谱设置中的行为不同于经典边设置，在经典边设置中，$q$ 的范围为 $n$ 阶，且对团由 Lovász 和 Simonovits 给出阈值为 $1/r$，对奇圈由 Pikhurko 和 Yilma 给出阈值为 $1/2$。
英文摘要
Let $Y_{n,r,q}$ be the graph obtained from the Turán graph $T_{n,r}$ by adding $q$ pairwise disjoint edges inside a largest part, and let $c(n,F)$ be the minimum number of copies of $F$ created by adding a single edge to $T_{n,r}$. Fang, Li, Lin and Ma proved that for every color-critical graph $F$ with $χ(F)=r+1$, there exists a constant $δ_F>0$ such that for all sufficiently large $n$ and all $1\le q\le δ_F \sqrt{n}$, the condition $λ(G)\geλ(Y_{n,r,q})$ forces at least $q\, c(n,F)$ copies of $F$. The bound $q=O(\sqrt{n}\,)$ is tight up to a constant factor, in contrast to the linear order $n$ of the edge setting of Mubayi, Pikhurko and Yilma, but the exact constant $δ_F$ remained unknown for any $F$. In this paper, building on a structural result of Fang, Li, Lin and Ma, we determine the threshold $δ_F$ when $F$ is a clique and an odd cycle. For every $r\ge2$, we denote $δ_r :=(1-\tfrac1r)\sqrt2$ and prove that for every $\varepsilon>0$, if $n$ is sufficiently large and $1\le q\le(δ_r-\varepsilon)\sqrt n$, then every $n$-vertex graph $G$ with $λ(G)\geλ(Y_{n,r,q})$ contains at least $q\,c(n,K_{r+1})$ copies of $K_{r+1}$, and $δ_r$ is best possible. For odd cycles, the threshold is $1/\sqrt2$. For every $k\ge1$ and $\varepsilon>0$, if $n$ is sufficiently large and $1\le q\le(1/\sqrt2-\varepsilon)\sqrt n$, then every $n$-vertex graph $G$ with $λ(G)\geλ(Y_{n,2,q})$ contains at least $q\,c(n,C_{2k+1})$ copies of $C_{2k+1}$, and $1/\sqrt2$ is best possible. Our results determine both the exact count of copies and the optimal range of $q$. The behavior in the spectral setting differs from the classical edge setting, in which the range of $q$ is of order $n$ and the threshold is $1/r$ for cliques by Lovász and Simonovits, and $1/2$ for odd cycles by Pikhurko and Yilma.
Comments24 pages, 1 table. Any comments are welcome
