Let $\mathcal{ F}$ be a family of $r$-graphs. The Tur\'an number $ex_r(n;\mathcal{ F})$ is defined to be the maximum number of edges in an $r$-graph of order $n$ that is $\mathcal{ F}$-free. The famous Erd\H{ o}s Matching Conjecture shows that \[ ex_r(n,M_{ k+1}^{ (r)})= \max\left\{ \binom{ rk+r-1}{ r},\binom{ n}{ r}-\binom{ n-k}{ r}\right\}, \] where $M_{ k+1}^{ (r)}$ represents the $r$-graph consisting of $k+1$ disjoint edges. Motivated by this conjecture, we consider the Tur\'an problem for tight linear forests. A tight linear forest is an $r$-graph whose connected components are all tight paths or isolated vertices. Let $\mathcal{ L}_{ n,k}^{ (r)}$ be the family of all tight linear forests of order $n$ with $k$ edges in $r$-graphs. In this paper, we prove that for sufficiently large $n$, \[ ex_r(n;\mathcal{ L}_{ n,k}^{ (r)})=\max\left\{ \binom{ k}{ r}, \binom{ n}{ r}-\binom{ n-\left\lfloor (k-1)/r\right \rfloor}{ r}\right\}+d, \] where $d=o(n^r)$ and if $r=3$ and $k=cn$ with $0<c<1$, if $r\geq 4$ and $k=cn$ with $0<c<1/2$. The proof is based on the weak regularity lemma for hypergraphs. We also conjecture that for arbitrary $k$ satisfying $k \equiv 1\ (mod\ r)$, the error term $d$ in the above result equals 0. We prove that the proposed conjecture implies the Erd\H{ o}s Matching Conjecture directly.
Categories: math.CO