On An Extremal Hypergraph Problem Of Brown, Erdős And Sós

Let f r (n,v,e) denote the maximum number of edges in an r-uniform hypergraph on n vertices, which does not contain e edges spanned by v vertices. Extending previous results of Ruzsa and Szemerédi and of Erdős, Frankl and Rödl, we partially resolve a problem raised by Brown, Erdős and Sós in 1973, by showing that for any fixed 2≤k<r, we have

$$ n^{{k - o{\left( 1 \right)}}} < f_{r} {\left( {n,3{\left( {r - k} \right)} + k + 1,3} \right)} = o{\left( {n^{k} } \right)}. $$

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Correspondence to Noga Alon*.

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* Researchs upported in part by a USA-Israeli BSF grant, by the Israel Science Foundation and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University.

† This work forms part of the author's Ph.D. Thesis. Research supported by a Charles Clore Foundation Fellowship and an IBM Ph.D. Fellowship.

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Alon*, N., Shapira†, A. On An Extremal Hypergraph Problem Of Brown, Erdős And Sós. Combinatorica 26, 627–645 (2006). https://doi.org/10.1007/s00493-006-0035-9

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Mathematics Subject Classification (2000):

  • 05C65
  • 05D99