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- W2560821812 abstract "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 $$ex(n, K^{(r)}_{s_1,s_2,cdots,s_r})=Thetaleft(n^{r-frac{1}{s_1s_2cdots s_{r-1}}}right).$$ 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 Verstraete independently with a different approach." @default.
- W2560821812 created "2016-12-16" @default.
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- W2560821812 date "2016-12-05" @default.
- W2560821812 modified "2023-09-23" @default.
- W2560821812 title "Some extremal results on complete degenerate hypergraphs" @default.
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- W2560821812 doi "https://doi.org/10.48550/arxiv.1612.01363" @default.
- W2560821812 hasPublicationYear "2016" @default.
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