TY - JOUR

T1 - A new bound for the Brown–Erdős–Sós problem

AU - Conlon, David

AU - Gishboliner, Lior

AU - Levanzov, Yevgeny

AU - Shapira, Asaf

N1 - Publisher Copyright:
© 2022 The Authors

PY - 2023/1

Y1 - 2023/1

N2 - Let f(n,v,e) denote the maximum number of edges in a 3-uniform hypergraph not containing e edges spanned by at most v vertices. One of the most influential open problems in extremal combinatorics then asks, for a given number of edges e≥3, what is the smallest integer d=d(e) such that f(n,e+d,e)=o(n2)? This question has its origins in work of Brown, Erdős and Sós from the early 70's and the standard conjecture is that d(e)=3 for every e≥3. The state of the art result regarding this problem was obtained in 2004 by Sárközy and Selkow, who showed that f(n,e+2+⌊log2e⌋,e)=o(n2). The only improvement over this result was a recent breakthrough of Solymosi and Solymosi, who improved the bound for d(10) from 5 to 4. We obtain the first asymptotic improvement over the Sárközy–Selkow bound, showing that f(n,e+O(loge/logloge),e)=o(n2).

AB - Let f(n,v,e) denote the maximum number of edges in a 3-uniform hypergraph not containing e edges spanned by at most v vertices. One of the most influential open problems in extremal combinatorics then asks, for a given number of edges e≥3, what is the smallest integer d=d(e) such that f(n,e+d,e)=o(n2)? This question has its origins in work of Brown, Erdős and Sós from the early 70's and the standard conjecture is that d(e)=3 for every e≥3. The state of the art result regarding this problem was obtained in 2004 by Sárközy and Selkow, who showed that f(n,e+2+⌊log2e⌋,e)=o(n2). The only improvement over this result was a recent breakthrough of Solymosi and Solymosi, who improved the bound for d(10) from 5 to 4. We obtain the first asymptotic improvement over the Sárközy–Selkow bound, showing that f(n,e+O(loge/logloge),e)=o(n2).

KW - Brown-Erdős-Sós conjecture

KW - Hypergraph removal lemma

UR - http://www.scopus.com/inward/record.url?scp=85137767185&partnerID=8YFLogxK

U2 - 10.1016/j.jctb.2022.08.005

DO - 10.1016/j.jctb.2022.08.005

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AN - SCOPUS:85137767185

SN - 0095-8956

VL - 158

SP - 1

EP - 35

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

ER -