TY - JOUR

T1 - Proof of the Brown–Erdös–Sós conjecture in groups

AU - Nenadov, Rajko

AU - Sudakov, Benny

AU - Tyomkyn, Mykhaylo

N1 - Publisher Copyright:
© Cambridge Philosophical Society 2019.

PY - 2020/9

Y1 - 2020/9

N2 - The conjecture of Brown, Erdös 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(n2) 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–Erdös–Sós conjecture for all finite groups and values of k. Moreover, we prove that the hypergraphs coming from groups contain sets of size Θ(√k) which span k edges. This is best possible and goes far beyond the conjecture.

AB - The conjecture of Brown, Erdös 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(n2) 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–Erdös–Sós conjecture for all finite groups and values of k. Moreover, we prove that the hypergraphs coming from groups contain sets of size Θ(√k) which span k edges. This is best possible and goes far beyond the conjecture.

UR - http://www.scopus.com/inward/record.url?scp=85068511863&partnerID=8YFLogxK

U2 - 10.1017/S0305004119000203

DO - 10.1017/S0305004119000203

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AN - SCOPUS:85068511863

SN - 0305-0041

VL - 169

SP - 323

EP - 333

JO - Mathematical Proceedings of the Cambridge Philosophical Society

JF - Mathematical Proceedings of the Cambridge Philosophical Society

IS - 2

ER -