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OP-9378OpenMathematics · general

Erdős #1158 · hypergraphs / turan number

Let Kt(r)K_{t}(r) be the complete tt-partite tt-uniform hypergraph with rr vertices in each class.

Is it true that

ext(n,Kt(r))≥nt−r1−t−o(1)\mathrm{ex}_t(n,K_t(r)) \geq n^{t-r^{1-t}-o(1)}

for all t,rt,r?

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

Let Kt(r)K_{t}(r) be the complete tt-partite tt-uniform hypergraph with rr vertices in each class.

Is it true that

ext(n,Kt(r))≥nt−r1−t−o(1)\mathrm{ex}_t(n,K_t(r)) \geq n^{t-r^{1-t}-o(1)}

for all t,rt,r?

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Background

Erdős [Er64f] proved that

nt−O(r1−t)≤ext(n,Kt(r))≪nt−r1−t.n^{t-O(r^{1-t})}\leq \mathrm{ex}_t(n,K_t(r)) \ll n^{t-r^{1-t}}.

This is only known when t=2t=2 and 2≤r≤32\leq r\leq 3. The case t=2t=2 is [714].

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 #1158 link

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