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OP-9334OpenMathematics · general

Erdős #1075 · hypergraphs

Let r≥3r\geq 3. There exists cr>r−rc_r>r^{-r} such that, for any ϵ>0\epsilon>0, if nn is sufficiently large, the following holds.

Any rr-uniform hypergraph on nn vertices with at least (1+ϵ)(n/r)r(1+\epsilon)(n/r)^r many edges cont…

Source reports OPENAs of 2026-09-08Source & evidence ↓
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Problem statement

Let r≥3r\geq 3. There exists cr>r−rc_r>r^{-r} such that, for any ϵ>0\epsilon>0, if nn is sufficiently large, the following holds.

Any rr-uniform hypergraph on nn vertices with at least (1+ϵ)(n/r)r(1+\epsilon)(n/r)^r many edges contains a subgraph on mm vertices with at least crmrc_rm^r edges, where m=m(n)→∞m=m(n)\to \infty as n→∞n\to \infty.

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Background

Erdős [Er64f] proved that this is true with cr=r−rc_r=r^{-r} whenever the graph has at least ϵnr\epsilon n^r many edges.

Progress & source assessment

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 #1075 link

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