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- W2953855682 abstract "Abstract The conjecture of Brown, Erdős and Sós from 1973 states that, for any k ≥ 3, if a 3-uniform hypergraph H with n vertices does not contain a set of k +3 vertices spanning at least k edges then it has o(n 2 ) edges. The case k = 3 of this conjecture is the celebrated (6, 3)-theorem of Ruzsa and Szemerédi which implies Roth’s theorem on 3-term arithmetic progressions in dense sets of integers. Solymosi observed that, in order to prove the conjecture, one can assume that H consists of triples (a, b, ab) of some finite quasigroup Γ. Since this problem remains open for all k ≥ 4, he further proposed to study triple systems coming from finite groups. In this case he proved that the conjecture holds also for k = 4. Here we completely resolve the Brown–Erdős–Sós conjecture for all finite groups and values of k . Moreover, we prove that the hypergraphs coming from groups contain sets of size $Theta (sqrt k )$ which span k edges. This is best possible and goes far beyond the conjecture." @default.
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- W2953855682 date "2019-07-04" @default.
- W2953855682 modified "2023-09-27" @default.
- W2953855682 title "Proof of the Brown–Erdős–Sós conjecture in groups" @default.
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- W2953855682 doi "https://doi.org/10.1017/s0305004119000203" @default.
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