AI 中文总结
该研究针对线性超图中生长冠的结构，证明缺陷接近零的边几乎全与高阶缺陷边相邻，得到不含指定冠的线性超图的渐近Turán系数上界。
AI 中文摘要
令$q=r-1$，$t=k-1$，且$D=tq+1$。对于不含$C^r_{1,k}$的线性$r$一致超图的一条边$e$，定义$δ_H(e)=\sum_{v\in e}\frac{1}{d_H(v)}-\frac{r}{D}$。该缺陷满足$δ_H(e)\ge 0$。取等时，$e$的每个顶点的度均为$D$，且$e$处的花瓣迹是$t$个$q$阶仿射平面的不交并。等价地，恢复基线后可得到以$e$为公共线的$t$个$q$阶射影平面。\n对于生长冠，等号结构在如下意义下是稳定的：若$q_j\to\infty$，$2\le t_j\le q_j$，且$e_j$是有限不含$C_{1,t_j+1}^{q_j+1}$的线性超图的一条边，满足$\frac{t_j^2}{q_j}\to0$、$t_j^3δ_{H_j}(e_j)\to0$，则对任意固定的$0<θ<1$，有$\frac{ |\{f\ne e_j:f\cap e_j\ne\varnothing,\\ δ_{H_j}(f)\geθ/t_j^2\}| }{(q_j+1)(t_jq_j)} \to1$。因此接近等号的边几乎完全与缺陷阶至少为$t_j^{-2}$的边相邻。一致形式给出绝对常数$Q_0,\varepsilon_0,c_0>0$，使得当$q\ge Q_0t^2$且$δ_H(e)<\varepsilon_0/t^3$时，$e$的至少$\frac12(q+1)tq$个邻边的缺陷至少为$1/(100t^2)$。由此可得$c^{\mathrm{lin}}_{q+1,t+1}\le t-\frac{c_0}{t}$，其中$c^{\mathrm{lin}}_{r,k}$表示$C^r_{1,k}$的渐近线性Turán系数。
英文摘要
Put $q=r-1$, $t=k-1$, and $D=tq+1$. For an edge $e$ of a linear $C^r_{1,k}$-free $r$-uniform hypergraph, define \[ δ_H(e)=\sum_{v\in e}\frac{1}{d_H(v)}-\frac{r}{D}. \] The defect satisfies $δ_H(e)\ge 0$. At equality, every vertex of $e$ has degree $D$, and the petal trace at $e$ is a disjoint union of $t$ affine planes of order $q$. Equivalently, restoring the base line gives $t$ projective planes of order $q$ with common line $e$. For growing crowns, the equality structure is stable in the following sense. If $q_j\to\infty$, $2\le t_j\le q_j$, and $e_j$ is an edge of a finite linear $C_{1,t_j+1}^{q_j+1}$-free hypergraph satisfying \[ \frac{t_j^2}{q_j}\to0, \qquad t_j^3δ_{H_j}(e_j)\to0, \] then, for every fixed $0<θ<1$, \[ \frac{ |\{f\ne e_j:f\cap e_j\ne\varnothing,\ δ_{H_j}(f)\geθ/t_j^2\}| }{(q_j+1)(t_jq_j)} \to1. \] Thus an edge close to equality is adjacent almost entirely to edges with defect of order at least $t_j^{-2}$. A uniform form gives absolute constants $Q_0,\varepsilon_0,c_0>0$ such that, whenever $q\ge Q_0t^2$ and $δ_H(e)<\varepsilon_0/t^3$, at least $\tfrac12(q+1)tq$ neighbors of $e$ have defect at least $1/(100t^2)$. Consequently, \[ c^{\mathrm{lin}}_{q+1,t+1}\le t-\frac{c_0}{t}, \] where $c^{\mathrm{lin}}_{r,k}$ denotes the asymptotic linear Turán coefficient for $C^r_{1,k}$.
Comments16 pages
