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Tur\'an numbers of theta graphs
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abstract
The theta graph $\Theta_{\ell,t}$ consists of two vertices joined by $t$ vertex-disjoint paths of length $\ell$ each. For fixed odd $\ell$ and large $t$, we show that the largest graph not containing $\Theta_{\ell,t}$ has at most $c_{\ell} t^{1-1/\ell}n^{1+1/\ell}$ edges and that this is tight apart from the value of $c_{\ell}$.
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3-uniform hypergraphs with few Berge paths of length three between any two vertices
For 3-uniform hypergraphs, the maximum number of edges in an n-vertex hypergraph with no Berge theta made of 217 internally disjoint length-3 paths is Omega(n^{4/3}), matching the upper bound up to a constant.
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