均匀超树的线性图兰数
Linear Turán Numbers of Uniform Hypertrees
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中文总结 AI 辅助
研究均匀超图族中线性图兰数，确定\(r\) - 均匀线性星\(S_k^r\)的线性图兰数，构造不含\(T_k^r\)的稠密超图，研究四条超边的线性超树，给出\(r = 4\)时\(ex^{\mathrm{lin}}_4(n,P_4^4)\)的新证明及相关结果。
中文摘要 AI 辅助
超图如果每对顶点至多包含在一条超边中，则为线性超图。对于\(r\) - 均匀超图族\(\mathcal{F}\)，线性图兰数\(ex^{\mathrm{lin}}_r(n,\mathcal{F})\)是\(n\)个顶点的不含\(\mathcal{F}\)的线性\(r\) - 均匀超图中超边的最大数量。扩展Gyárfás、Ruszinkó和Sárközy关于\(3\) - 均匀线性超树的工作，研究更高均匀性的线性图兰数。确定了\(r\) - 均匀线性星\(S_k^r\)的线性图兰数，证明\([ex^{\mathrm{lin}}_r(n,S_k^r)\le \frac{n(k - 1)}{r}]\)，等号成立当且仅当存在\((k - 1)\) - 正则线性\(r\) - 均匀超图。还构造了不含\(T_k^r\)的稠密超图，表明在合适的整除性和设计存在假设下，对于每个有\(k\)条超边的线性\(r\) - 均匀超树\(T_k^r\)，\([ex^{\mathrm{lin}}_r(n,T_k^r)\ge \frac{n(k - 1)}{r}]\)。然后研究了所有有四条超边的线性超树，对于扫帚\(B_4^r\)，证明\([ex^{\mathrm{lin}}_r(n,B_4^r)\le \frac{(r + 1)n}{r}]\)，并将极值超图刻画为Steiner系统\(S(2,r,r^2)\)的不相交并集（只要存在）。对于皇冠\(E_4^r\)，确定\([ex^{\mathrm{lin}}_r(n,E_4^r)\le \frac{(2r - 1)n}{r}]\)，并给出一个仅留下常数因子差距的下界构造。对于线性路径\(P_4^r\)，构造了具有\((r + 1)n/r\)条超边的不含\(P_4^r\)的超图，并猜想这是最优的一般界，在合适的度条件下验证了连通超图的猜想。最后，对于\(r = 4\)，指出Zhang和Wang先前证明中的一个关键结构断言的反例，给出\([ex^{\mathrm{lin}}_4(n,P_4^4)\le \frac{5n}{4}]\)的新证明，并表明等号成立当且仅当是Steiner系统\(S(2,4,16)\)的不相交并集。
英文摘要
A hypergraph is \emph{linear} if every pair of vertices is contained in at most one hyperedge. For a family $\mathcal{F}$ of $r$-uniform hypergraphs, let $\operatorname{ex}^{\mathrm{lin}}_r(n,\mathcal{F})$ denote the maximum number of hyperedges in an $n$-vertex $\mathcal{F}$-free linear $r$-uniform hypergraph. Extending earlier work on acyclic triple systems, we study linear Turán numbers of uniform hypertrees in higher uniformity. For the linear star $S_k^r$, we prove \[ \operatorname{ex}^{\mathrm{lin}}_r(n,S_k^r)\leq \frac{n(k-1)}{r}, \] with equality precisely for $(k-1)$-regular linear $r$-uniform hypergraphs, whenever such hypergraphs exist. Under suitable divisibility and design-existence assumptions, we also construct $T_k^r$-free hypergraphs with $n(k-1)/r$ edges for every linear $r$-uniform hypertree $T_k^r$ with $k$ hyperedges. For the four-edge broom $B_4^r$, we prove \[ \operatorname{ex}^{\mathrm{lin}}_r(n,B_4^r)\leq \frac{(r+1)n}{r}, \] with equality exactly for disjoint unions of Steiner systems $S(2,r,r^2)$, whenever such systems exist. For the crown $E_4^r$, we establish a degree-sensitive upper bound implying \[ \operatorname{ex}^{\mathrm{lin}}_r(n,E_4^r)\leq \frac{(2r-1)n}{r}, \] and give a lower-bound construction leaving a constant-factor gap. Finally, we settle the linear Turán problem for the four-edge path $P_4^r$ in every uniformity: \[ \operatorname{ex}^{\mathrm{lin}}_r(n,P_4^r)\leq \frac{(r+1)n}{r}. \] Equality holds precisely for disjoint unions of Steiner systems $S(2,r,r^2)$. We also give counterexamples to a key structural claim used in a previously proposed proof of the $4$-uniform case.
