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OP-8855OpenCombinatorics

Erdős #142 · additive combinatorics / arithmetic progressions

Let rk(N)r_k(N) be the largest possible size of a subset of {1,…,N}\{1,\ldots,N\} that does not contain any non-trivial kk-term arithmetic progression. Prove an asymptotic formula for rk(N)r_k(N).

MSC 05 · Combinatorics

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

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

Let rk(N)r_k(N) be the largest possible size of a subset of {1,…,N}\{1,\ldots,N\} that does not contain any non-trivial kk-term arithmetic progression. Prove an asymptotic formula for rk(N)r_k(N).

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Background

Erdős remarked this is 'probably unattackable at present'. In [Er97c] Erdős offered $1000, but given that he elsewhere offered $5000 just for (essentially) showing that rk(N)=ok(N/log⁡N)r_k(N)=o_k(N/\log N) (see [3]), that value seems odd. In [Er81] he offers $10000, stating it is 'probably enormously difficult'.

The best known upper bounds for rk(N)r_k(N) are due to Kelley and Meka [KeMe23] for k=3k=3, Green and Tao [GrTa17] for k=4k=4, and Leng, Sah, and Sawhney [LSS24] for k≥5k\geq 5. An asymptotic formula is still far out of reach, even for k=3k=3.

In [Er80] and [Va99] he asks (much more reasonably) for the order of magnitude of rk(N)r_k(N). In [Er80] he remarks that we do not even know whether rk(n)/rk+1(n)→0r_k(n)/r_{k+1}(n)\to 0 for any k≥3k\geq 3.

See also [3] and [139].

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 #142 link
  2. [2]Upstream Lean 4 statement for Erdős #142 linkExternal statement reference. Not compiled or verified by this platform; upstream files may contain open proof obligations.
  3. [3]Formal Conjectures — Erdős Problem 142 link

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