A New Bound for the Brown-Erdős-Sós Problem

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) so 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)

A New Bound for the Brown--Erd\H{o}s--S\'os Problem

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 $e \geq 3$, what is the smallest integer $d=d(e)$ so that $f(n,e+d,e) = o(n^2)$? This question has its origins in work of Brown, Erd\H{o}s and S\'os from the early 70's and the standard conjecture is that $d(e)=3$ for every $e \geq 3$. The state of the art result regarding this problem was obtained in 2004 by S\'{a}rk\"{o}zy and Selkow, who showed that $f(n,e + 2 + \lfloor \log_2 e \rfloor,e) = o(n^2)$. 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 S\'{a}rk\"{o}zy--Selkow bound, showing that $$ f(n, e + O(\log e/ \log\log e), e) = o(n^2). $

A new bound for the Brown–Erdős–Sós problem

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).ISSN:0095-895