偶环的悬挂图的极值问题
Extremal problems for suspensions of even cycles
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中文总结 AI 辅助
本文研究偶环悬挂图的两类极值问题，确定了特定参数下 $k$ 均匀超图中不含对应悬挂图时边数、完全图副本数的极值量级，推广了经典图论结论并拓展至单纯复形 Turán 问题。
中文摘要 AI 辅助
给定整数 $k\geq2$ 和图 $F$，$k$ 均匀悬挂图 $\mathcal{S}^kF$ 是通过向 $F$ 的每条边添加固定的 $k-2$ 个新顶点得到的。本文研究偶环的悬挂图的两个极值问题。记 $K^k_t$ 为 $t$ 阶 $k$ 均匀完全图，$\mathrm{ex}(n,\mathcal{S}^kC_{2\ell})$ 和 $\mathrm{ex}(n,K^k_t,\mathcal{S}^kC_{2\ell})$ 分别表示不含 $\mathcal{S}^kC_{2\ell}$ 的 $n$ 顶点 $k$ 均匀超图中边的最大数量和 $K^k_t$ 副本的最大数量。我们证明，对任意 $k\geq2$ 及无穷多 $n$，有 $\mathrm{ex}(n,K^{k}_{k+1},\mathcal{S}^kC_4)=\frac{n^{k-1/2}}{(k+1)!}+O(n^{k-1})$，该结果推广了 $k=2$ 时的经典结论，且作为直接推论得到了无穷多 $n$ 时 $\mathrm{ex}(n,\mathcal{S}^kC_4)$ 的渐近值（此前 Mubayi 已对所有 $n$ 证明该结论）。此外，对任意 $k\geq2$ 及 $\ell\in\{3,5\}$，我们确定 $\mathrm{ex}(n,\mathcal{S}^kC_{2\ell})=\Theta(n^{k-1+1/\ell})$ 的量级，该结果推广了 $k=2$ 时的经典图情形及 Mukherjee 关于 $k=\ell=3$ 的先前结果。两个问题的主要难点在于下界构造：我们对 $\mathrm{ex}(n,K^{k}_{k+1},\mathcal{S}^kC_4)$ 的构造采用了新颖的块填充结构，其生成的 $K^k_{k+1}$ 副本数量远多于已知构造；对 $\ell\in\{3,5\}$ 的 $\mathrm{ex}(n,\mathcal{S}^kC_{2\ell})$，我们建立了自然的 $k$ 均匀版本的 Wenger 图，解决了将极值图构造提升至悬挂图时的微妙问题，还将结果应用于单纯复形的 Turán 问题。
英文摘要
Given an integer $k\geq2$ and a graph $F$, the $k$-uniform suspension $\mathcal{S}^kF$ is obtained by adjoining a fixed set of $k-2$ new vertices to every edge of $F$. In this paper, we study two extremal problems for suspensions of even cycles. Write $K^k_t$ for the $k$-uniform clique of order $t$. Let $\mathrm{ex}(n,\mathcal{S}^kC_{2\ell})$ and $\mathrm{ex}(n,K^k_t,\mathcal{S}^kC_{2\ell})$ denote the maximum numbers of edges and copies of $K^k_t$, respectively, in an $\mathcal{S}^kC_{2\ell}$-free $k$-uniform hypergraph on $n$ vertices. We prove that, for every $k\geq2$ and infinitely many $n$, \[\mathrm{ex}(n,K^{k}_{k+1},\mathcal{S}^kC_4)=\frac{n^{k-1/2}}{(k+1)!}+O(n^{k-1}).\] This extends a folklore result for $k=2$ and, as an immediate consequence, yields the asymptotics of $\mathrm{ex}(n,\mathcal{S}^kC_4)$ for infinitely many $n$, previously established by Mubayi (for all $n$). Furthermore, for every $k\geq2$ and $\ell\in\{3,5\}$, we determine the order of magnitude \[\mathrm{ex}(n,\mathcal{S}^kC_{2\ell})=Θ(n^{k-1+1/\ell}).\] This generalizes both the classical graph case $k=2$ and a previous result of Mukherjee for $k=\ell=3$. The principal difficulty in both problems lies in constructing the lower bounds. Our construction for $\mathrm{ex}(n,K^{k}_{k+1},\mathcal{S}^kC_4)$ incorporates a novel block-packing structure, which yields substantially more copies of $K^k_{k+1}$ than previously known constructions. For $\mathrm{ex}(n,\mathcal{S}^kC_{2\ell})$ with $\ell\in\{3,5\}$, we establish a natural $k$-uniform version of Wenger graphs, addressing the subtleties involved in lifting extremal graph constructions to suspensions. We also give applications of our results to Turán problems for simplicial complexes.
