Some tight lower bounds for Tur\'{a}n problems via constructions of multi-hypergraphs

Recently, several hypergraph Tur'{a}n problems were solved by the powerful random algebraic method. However, the random algebraic method usually requires some parameters to be very large, hence we are concerned about how these Tur'{a}n numbers depend on such large parameters of the forbidden hypergraphs. In this paper, we determine the dependence on such specified large constant for several hypergraph Tur'{a}n problems. More specifically, for complete $r$-partite $r$-uniform hypergraphs, we show that if $s_r$ is sufficiently larger than $s_1,s_2,\dots ,s_{r-1},$ then $${\text{ex}}_r(n,K_{s_1,s_2,\dots ,s_r}^{(r)})=\mathrm{\Theta }(s_r^{\frac{1}{s_1{s}_2\cdots s_{r-1}}}{n}^{r-\frac{1}{s_1{s}_2\cdots s_{r-1}}})\mathrm{.}$$ For complete bipartite $r$-uniform hypergraphs, we prove that if $s$ is sufficiently larger than $t,$ we have $${\text{ex}}_r(n,K_{s,t}^{(r)})=\mathrm{\Theta }(s^{\frac{1}{t}}{n}^{r-\frac{1}{t}})\mathrm{.}$$ In particular, our results imply that the famous K\H{o}v'{a}ri–S'{o}s–Tur'{a}n’s upper bound $\text{ex}(n,K_{s,t})=O(t^{\frac{1}{s}}{n}^{2-\frac{1}{s}})$ has the correct dependence on large $t$. The main approach is to construct random multi-hypergraph via a variant of random algebraic method.