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Graphs with few paths of prescribed length between any two vertices
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abstract
We use a variant of Bukh's random algebraic method to show that for every natural number $k \geq 2$ there exists a natural number $\ell$ such that, for every $n$, there is a graph with $n$ vertices and $\Omega_k(n^{1 + 1/k})$ edges with at most $\ell$ paths of length $k$ between any two vertices. A result of Faudree and Simonovits shows that the bound on the number of edges is tight up to the implied constant.
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Cited by 1 Pith paper
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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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