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Erdős #1178 · graph theory / hypergraphs

For r≥3r\geq 3 let dr(e)d_r(e) be the minimal dd such that

exr(n,F)=o(n2),\mathrm{ex}_r(n,\mathcal{F})=o(n^2),

where F\mathcal{F} is the family of rr-uniform hypergraphs on dd vertices with ee edges.

Prove that

dr(e)=(r−…d_r(e)=(r-…
MSC 05C · Graph theory

Inferred from the community field; finer classification has not been reviewed.

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Problem statement

For r≥3r\geq 3 let dr(e)d_r(e) be the minimal dd such that

exr(n,F)=o(n2),\mathrm{ex}_r(n,\mathcal{F})=o(n^2),

where F\mathcal{F} is the family of rr-uniform hypergraphs on dd vertices with ee edges.

Prove that

dr(e)=(r−2)e+3d_r(e)=(r-2)e+3

for all r,e≥3r,e\geq 3.

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Background

A conjecture of Brown, Erdős, and Sós [BES73], who proved that dr(e)≥(r−2)e+3d_r(e)\geq (r-2)e+3 (see also [1076]).

Ruzsa and Szemerédi [RuSz78] proved d3(3)=6d_3(3)=6 (see [716]).

Erdős, Frankl, and Rödl [EFR86] proved

dr(3)=(r−2)3+3d_r(3)= (r-2)3+3

for all r≥3r\geq 3. Sárközy and Selkow [SaSe05] proved

dr(e)≤(r−2)e+2+⌊log⁡2e⌋d_r(e) \leq (r-2)e+2+\lfloor \log_2 e\rfloor

for all r,e≥3r,e\geq 3. Solymosi and Solymosi [SoSo17] proved that d3(10)≤14d_3(10)\leq 14. Conlon, Gishboliner, Levanzov, and Shapira [CGLS23] proved

d3(e)≤e+O(log⁡elog⁡log⁡e)d_3(e)\leq e+O\left(\frac{\log e}{\log\log e}\right)

for all e≥3e\geq 3.

In [Er75b] Erdős even asks whether, if F\mathcal{F} is the family of 33-uniform hypergraphs on kk vertices with k−3k-3 edges,

ex3(n,F)≍nrk−3(n),\mathrm{ex}_3(n,\mathcal{F})\asymp n r_{k-3}(n),

where rk−3(n)r_{k-3}(n) is the maximal size of a subset of {1,…,n}\{1,\ldots,n\} that does not contain a non-trivial arithmetic progression of length k−3k-3. He states that Ruzsa has proved the lower bound for k=6,7,8k=6,7,8.

See [1157] for the general case.

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The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #1178 link

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