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 $s_{ 1},s_{ 2},\ldots,s_{ r-1},$ then $$\textup{ ex}_{ r}(n,K_{ s_{ 1},s_{ 2},\ldots,s_{ r}}^{ (r)})=\Theta(s_{ r}^{ \frac{ 1}{ s_{ 1}s_{ 2}\cdots s_{ r-1}}}n^{ r-\frac{ 1}{ s_{ 1}s_{ 2}\cdots s_{ r-1}}}).$$ For complete bipartite $r$-uniform hypergraphs, we prove that if $s$ is sufficiently larger than $t,$ we have $$\textup{ ex}_{ r}(n,K_{ s,t}^{ (r)})=\Theta(s^{ \frac{ 1}{ t}}n^{ r-\frac{ 1}{ t}}).$$ In particular, our results imply that the famous K\H{ o}v\'{ a}ri--S\'{ o}s--Tur\'{ a}n's upper bound $\textup{ ex}(n,K_{ s,t})=O(t^{ \frac{ 1}{ s}}n^{ 2-\frac{ 1}{ s}})$ has the correct dependence on large $t$. The main approach is to construct random multi-hypergraph via a variant of random algebraic method.
Categories: math.CO