Some extremal results on hypergraph Tur\'{a}n problems

For two $r$-graphs $\mathcal{T}$ and $\mathcal{H}$, let ${\text{ex}}_r(n,\mathcal{T},\mathcal{H})$ be the maximum number of copies of $\mathcal{T}$ in an $n$-vertex $\mathcal{H}$-free $r$-graph. The determination of Tur'{a}n number ${\text{ex}}_r(n,\mathcal{T},\mathcal{H})$ has become the fundamental core problem in extremal graph theory ever since the pioneering work Tur'{a}n’s Theorem was published in $1941$. Although we have some rich results for the simple graph case, only sporadic results have been known for the hypergraph Tur'{a}n problems. In this paper, we mainly focus on the function ${\text{ex}}_r(n,\mathcal{T},\mathcal{H})$ when $\mathcal{H}$ is one of two different hypergraph extensions of the complete bipartite graph $K_{s,t}$. The first extension is the complete bipartite $r$-graph $K_{s,t}^{(r)}$, which was introduced by Mubayi and Verstra"{e}te~[J. Combin. Theory Ser. A, 106: 237–253, 2004]. Using the powerful random algebraic method, we show that if $s$ is sufficiently larger than $t$, then $${\text{ex}}_r(n,\mathcal{T},K_{s,t}^{(r)})=\mathrm{\Omega }(n^{v-\frac{e}{t}}),$$ where $\mathcal{T}$ is an $r$-graph with $v$ vertices and $e$ edges. In particular, when $\mathcal{T}$ is an edge or some specified complete bipartite $r$-graph, we can determine their asymptotics. The second important extension is the complete $r$-partite $r$-graph $K_{s_1,s_2,\dots ,s_r}^{(r)}$, which has been widely studied. When $r=3$, we provide an explicit construction giving $${\text{ex}}_3(n,K_{2,2,7}^{(3)})⩾\frac{1}{27}{n}^{\frac{19}{7}}+o(n^{\frac{19}{7}})\mathrm{.}$$ Our construction is based on the Norm graph, and improves the lower bound $\mathrm{\Omega }(n^{\frac{73}{27}})$ obtained by probabilistic method.