论文标题
线性turán数量的无环四倍系统
Linear Turán numbers of acyclic quadruple systems
论文作者
论文摘要
线性$ r $均匀的超图被称为acycilc如果可以从一个单个边缘开始构造，则在每个步骤中添加一个新的边缘，该新边缘与以前边缘的顶点的结合，最多是一个顶点。最近，Gyárfás，Ruszinkó和Sárk\''{O} Zy启动了线性Turán数量的无环线性三重系统的研究。在本文中，我们将其结果扩展到线性四倍系统。在这里，我们专注于小树，路径和匹配。对于小树的情况，我们发现，对于线性树$ t $，$ ex^{lin} _ {4}（n，t）$与Steiner System $ s（2,4，N）$上的困难问题有关，例如，我们表明$ ex^{lin} _ {4} _ {4}（n，p_4）（n，p_4）\ le \ frac ifrac ifrac ifrac ifrac ifrac ifrac ifrac ifrac ifrac ifrac ifrac {4 ifrac {5n}四连销系统是$ s（2,4,16）$的不连接联盟。用$ e^{+} _ {4} $表示线性树，由三个成对的脱节四倍体组成，第四个相交。我们证明$ 12 \ lfloor \ frac {n-4} {9} \ rfloor \ le ex^{lin} _ {4} _ {4}（N，E^{+} _ 4） $ p_k $分别具有$ k $成对的脱节四倍体和带有$ k $ QUADRUPLES的线性路径。对于路径的情况，我们表明$ ex^{lin} _ {4}（n，p_k）\ le 2.5kn $。 For the case of matchings, we prove that for fixed $k$ and sufficiently large $n$, $ex^{lin}_{4}(n, M_k)=g(n,k)$ where $g(n,k$) denotes the maximum number of quadruples that can intersect $k-1$ vertices in a linear quadruple system on $n$ vertices.
A linear $r$-uniform hypergraph is called acycilc if it can be constructed starting from one single edge then at each step adding a new edge that intersect the union of the vertices of the previous edges in at most one vertex. Recently, Gyárfás, Ruszinkó and Sárk\''{o}zy initiated the study of the linear Turán numbers of acyclic linear triple systems. In this paper, we extend their results to linear quadruple systems. Here, we concentrate on small trees, paths and matchings. For the case of small trees, we find that for a linear tree $T$, $ex^{lin}_{4}(n,T)$ relates to difficult problems on Steiner system $S(2,4,n)$ For example, we show that $ex^{lin}_{4}(n, P_4)\le \frac{5n}{4}$ with equality holds if and only if the linear quadruple system is the disjoint union of $S(2,4,16)$. Denote by $E^{+}_{4}$ the linear tree consisting of three pairwise disjoint quadruples and a fourth one intersecting all of them. We prove that $12\lfloor\frac{n-4}{9}\rfloor\le ex^{lin}_{4}(n, E^{+}_4)\le \frac{14(n-s)}{9}$, where $s$ is the number of vertices in $G$ with degree at least 8. Denote by $M_k$ and $P_k$ the set of $k$ pairwise disjoint quadruples and the linear path with $k$ quadruples, respectively. For the case of paths, we show that $ex^{lin}_{4}(n, P_k)\le 2.5kn$. For the case of matchings, we prove that for fixed $k$ and sufficiently large $n$, $ex^{lin}_{4}(n, M_k)=g(n,k)$ where $g(n,k$) denotes the maximum number of quadruples that can intersect $k-1$ vertices in a linear quadruple system on $n$ vertices.
