Some tight lower bounds for Turán problems via constructions of multi-hypergraphs

Recently, several hypergraph Turán 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 Turán 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 Turán 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 $\textup{ex}_{r}(n,K_{s_{1},s_{2},\ldots,s_{r}}^{(r)})=Θ(s_{r}^{\frac{1}{s_{1}s_{2}\cdots s_{r-1}}}n^{r-\frac{1}{s_{1}s_{2}\cdots s_{r-1}}}).$ For complete bipartite $r$-uniform hypergraphs, we prove that if $s$ is sufficiently larger than $t,$ we have $\textup{ex}_{r}(n,K_{s,t}^{(r)})=Θ(s^{\frac{1}{t}}n^{r-\frac{1}{t}}).$ In particular, our results imply that the famous Kővári--Sós--Turán'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.

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