Linear Tur\'an Numbers of Uniform Hypertrees

Abstract

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 Tur\'an 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 Tur\'an 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.