Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

15
The categories of this paper are: nucl-ex
27
The categories of this paper are: comp-gas
5
The categories of this paper are: gr-qc
19
The categories of this paper are: eess
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The categories of this paper are: cmp-lg
10
The categories of this paper are: q-bio
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The categories of this paper are: hep-ex
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The categories of this paper are: adap-org
24
The categories of this paper are: dg-ga
1
The categories of this paper are: cs
22
The categories of this paper are: chao-dyn
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The categories of this paper are: patt-sol
13
The categories of this paper are: quant-ph
16
The categories of this paper are: stat
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The categories of this paper are: cond-mat
3
The categories of this paper are: physics