SUMMARY
An rr-uniform hypergraph is linear if every two edges intersect in at most one vertex. Given a family of rr-uniform hypergraphs F\mathcal{F}, the linear Turán number exlinr(n,F)_r^{lin}(n,\mathcal{F}) is the maximum number of edges of a linear rr-uniform hypergraph on nn vertices that does not contain any member of F\mathcal{F} as a subgraph.Let KlK_l be a complete graph with ll vertices and r=2r\geq 2. The rr-expansion of KlK_l is the rr-graph K+lK_l^+ obtained from KlK_l by enlarging each edge of KlK_l with r-2r-2 new vertices disjoint from V(Kl)V(K_l) such that distinct edges of KlK_l are enlarged by distinct vertices. When l=r=3l\geq r \geq 3 and nn is sufficiently large, we prove the following extension of Turán's Theorem exlinr(n,K+l+1)=|TDr(n,l)|,ex_{r}^{lin}\left(n, K_{l+1}^{+}\right)\leq |TD_r(n,l)|, with equality achieved only by the Turán design TDr(n,l)TD_r(n,l), where the Turán design TDr(n,l)TD_r(n,l) is an almost balanced ll-partite rr-graph such that each pair of vertices from distinct parts are contained in one edge exactly. Moreover, some results on linear Turán number of general configurations are also presented.