Linear Tur\'{a}n numbers of acyclic quadruple systems

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, Gy\'{a}rf\'{a}s, Ruszink\'{o} and S\'{a}rk\''{o}zy initiated the study of the linear Tur\'{a}n 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.