Exact solution of the hypergraph Tur\'an problem for $k$-uniform linear paths

A $k$-uniform linear path of length $\mathrm{\ell }$, denoted by $P_{\mathrm{\ell }}^{(k)}$, is a family of $k$-sets $\{F_1,\mathrm{.}\mathrm{.}\mathrm{.},F_{\mathrm{\ell }}\}$ such that $\mathrm{\mid }{F}_i\cap F_{i+1}\mathrm{\mid }=1$ for each $i$ and $F_i\cap F_j=\mathrm{\varnothing }$ whenever $\mathrm{\mid }i-j\mathrm{\mid }>1$. Given a $k$-uniform hypergraph $H$ and a positive integer $n$, the {\it $k$-uniform hypergraph Tur'an number} of $H$, denoted by ${\text{ex}}_k(n,H)$, is the maximum number of edges in a $k$-uniform hypergraph $\text{cF}$ on $n$ vertices that does not contain $H$ as a subhypergraph. With an intensive use of the delta-system method, we determine ${\text{ex}}_k(n,P_{\mathrm{\ell }}^{(k)})$ exactly for all fixed $\mathrm{\ell }\ge 1,k\ge 4$, and sufficiently large $n$. We show that $${\text{ex}}_k(n,P_{2t+1}^{(k)})=\left(\genfrac{}{}{0px}{}{n-1}{k-1}\right)+\left(\genfrac{}{}{0px}{}{n-2}{k-1}\right)+\mathrm{.}\mathrm{.}\mathrm{.}+\left(\genfrac{}{}{0px}{}{n-t}{k-1}\right)\mathrm{.}$$ The only extremal family consists of all the $k$-sets in $[n]$ that meet some fixed set of $t$ vertices. We also show that $$\text{ex}(n,P_{2t+2}^{(k)})=\left(\genfrac{}{}{0px}{}{n-1}{k-1}\right)+\left(\genfrac{}{}{0px}{}{n-2}{k-1}\right)+\mathrm{.}\mathrm{.}\mathrm{.}+\left(\genfrac{}{}{0px}{}{n-t}{k-1}\right)+\left(\genfrac{}{}{0px}{}{n-t-2}{k-2}\right),$$ and describe the unique extremal family. Stability results on these bounds and some related results are also established.