Some extremal results on complete degenerate hypergraphs

Let $K^{(r)}_{s_1,s_2,\cdots,s_r}$ be the complete $r$-partite $r$-uniform hypergraph and $ex(n,K^{(r)}_{s_1,s_2,\cdots,s_r})$ be the maximum number of edges in any $n$-vertex $K^{(r)}_{s_1,s_2,\cdots,s_r}$-free $r$-uniform hypergraph. It is well-known in the graph case that $ex(n,K_{s,t})=\Theta(n^{2-1/s})$ when $t$ is sufficiently larger than $s$. In this note, we generalize the above to hypergraphs by showing that if $s_r$ is sufficiently larger than $s_1,s_2,\cdots,s_{r-1}$ then $$ex(n, K^{(r)}_{s_1,s_2,\cdots,s_r})=\Theta\left(n^{r-\frac{1}{s_1s_2\cdots s_{r-1}}}\right).$$ This follows from a more general Tur\'an type result we establish in hypergraphs, which also improves and generalizes some recent results of Alon and Shikhelman. The lower bounds of our results are obtained by the powerful random algebraic method of Bukh. Another new, perhaps unsurprising insight which we provide here is that one can also use the random algebraic method to construct non-degenerate (hyper-)graphs for various Tur\'an type problems. The asymptotics for $ex(n, K^{(r)}_{s_1,s_2,\cdots,s_r})$ is also proved by Verstra\"ete independently with a different approach.