每个图最终都是Turán-优的和Turán-稳定的

中文摘要 AI 辅助

设$T_r(n)$表示在$n$个顶点上各部分大小至多相差1的完全$r$-部图。若对于所有足够大的$n$，在所有$n$个顶点的$K_{r+1}$-自由图中，图$T_r(n)$包含$H$的副本数最多，则称图$H$是$K_{r+1}$-Turán-优的。若对于每个$\varepsilon>0$，存在$\delta>0$和$n_0$，使得当$n\ge n_0$且一个$n$个顶点的$K_{r+1}$-自由图$G$包含至少$\mathrm{ex}(n,H,K_{r+1})-\delta n^{v(H)}$个$H$的副本时，$G$与$T_r(n)$之间的编辑距离至多为$\varepsilon n^2$，其中$v(H)$表示$H$的阶，则称$H$是$K_{r+1}$-Turán-稳定的。在本文中，我们证明对于$r\ge 4v(H)^3+11v(H)^2$，每个图$H$既是$K_{r+1}$-Turán-优的又是$K_{r+1}$-Turán-稳定的。这加强了Morrison、Nir、Norin、Rzążewski和Wesolek~[J. Combin. Theory Ser. B, 2023]以及Gerbner和Hama Karim~[J. Graph Theory, 2024]的结果，并对Morrison、Nir、Norin、Rzążewski和Wesolek的一个问题给出了肯定回答。我们还证明，对于每个$n$个顶点的$K_{r+1}$-自由图$G$且$r\ge 40v(H)^3$，有$\inj(H,G)\le \inj(H,T_r(n))$，其中$\inj(H,G)$表示从$H$到$G$的单射同态数量。最后，我们对Morrison、Nir、Norin、Rzążewski和Wesolek的另一个问题给出了否定回答。

英文摘要

Let $T_r(n)$ denote the complete $r$-partite graph on $n$ vertices whose part sizes differ by at most one. A graph $H$ is called $K_{r+1}$-Turán-good if, for all sufficiently large $n$, the graph $T_r(n)$ contains the maximum number of copies of $H$ among all $n$-vertex $K_{r+1}$-free graphs. We say that $H$ is $K_{r+1}$-Turán-stable if, for every $\varepsilon>0$, there exist $δ>0$ and $n_0$ such that, whenever $n\ge n_0$ and an $n$-vertex $K_{r+1}$-free graph $G$ contains at least $\mathrm{ex}(n,H,K_{r+1})-δn^{v(H)}$ copies of $H$, the edit distance between $G$ and $T_r(n)$ is at most $\varepsilon n^2$, where $v(H)$ denotes the order of $H$. In this paper, we prove that every graph $H$ is both $K_{r+1}$-Turán-good and $K_{r+1}$-Turán-stable for $r\ge 4v(H)^3+11v(H)^2$. This strengthens results of Morrison, Nir, Norin, Rzążewski, and Wesolek~[J. Combin. Theory Ser. B, 2023] and Gerbner and Hama Karim~[J. Graph Theory, 2024], and gives a positive answer to a question of Morrison, Nir, Norin, Rzążewski, and Wesolek. We also prove that $\inj(H,G)\le \inj(H,T_r(n))$ for every $n$-vertex $K_{r+1}$-free graph $G$ and for $r\ge 40v(H)^3$, where $\inj(H,G)$ denotes the number of injective homomorphisms from $H$ to $G$. Finally, we give a negative answer to another question of Morrison, Nir, Norin, Rzążewski, and Wesolek.

发表机构

• School of Mathematics, Nanjing University（南京大学数学系）

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21 pages