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OP-9377OpenMathematics · general

Erdős #1157 · hypergraphs / turan number

Let t,k,r≥2t,k,r\geq 2. Let F\mathcal{F} be the family of all rr-uniform hypergraphs with kk vertices and ss edges. Determine

exr(n,F).\mathrm{ex}_r(n,\mathcal{F}).
Source reports OPENAs of 2026-09-08Source & evidence ↓
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Problem statement

Let t,k,r≥2t,k,r\geq 2. Let F\mathcal{F} be the family of all rr-uniform hypergraphs with kk vertices and ss edges. Determine

exr(n,F).\mathrm{ex}_r(n,\mathcal{F}).

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Background

This is a very difficult and general question, and many partial results are known. The paper of Brown, Erdős, and Sós [BES73] contains many of them, in particular the general lower bound, for all k>rk>r and s>1s>1,

exr(n,F)≫k,snrs−ks−1.\mathrm{ex}_r(n,\mathcal{F})\gg_{k,s} n^{\frac{rs-k}{s-1}}.

A general conjecture of Brown, Erdős, and Sós is that, for all r>t≥2r>t\geq 2 and s≥3s\geq 3,

ext(n,F)=o(nt)\mathrm{ex}_t(n,\mathcal{F})=o(n^t)

whenever k≥(r−t)s+t+1k\geq (r-t)s+t+1. The case t=2t=2 is the subject of [1178].

The case s=r=3s=r=3 and k=6k=6 is the subject of [716]. The case of r=3r=3 and k=s+2k=s+2 is the subject of [1076].

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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 #1157 link

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