Linear Turán number ex_r^lin(n,T_k^r) is at least n(k-1)/r for any r-uniform tree with k edges; exact upper bound (r+1)n/r for B_4^r with characterization, (2r-1)n/r for E_4^r, and matching lower construction for P_4^r.
Turán numbers for hypergraph star forests
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Bounds on Linear Tur\'{a}n Number for Trees
Linear Turán number ex_r^lin(n,T_k^r) is at least n(k-1)/r for any r-uniform tree with k edges; exact upper bound (r+1)n/r for B_4^r with characterization, (2r-1)n/r for E_4^r, and matching lower construction for P_4^r.