The Turán problem for a family of tight linear forests

Abstract: 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 $M_{k+1}<sup>{(r)}$ 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 $\mathcal{L}<em>{n,k}<sup>{(r)}$ 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 $d=o(n<sup>r)$ 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.