OpenFrontiers
Join
← Back to the index
OP-8877OpenCombinatorics

Erdős #176 · additive combinatorics / arithmetic progressions

Let N(k,ℓ)N(k,\ell) be the minimal NN such that for any f:{1,…,N}→{−1,1}f:\{1,\ldots,N\}\to\{-1,1\} there must exist a kk-term arithmetic progression PP such that

∣∑n∈Pf(n)∣≥ℓ.\left\lvert \sum_{n\in P}f(n)\right\rvert\geq \ell.

Find good up…

MSC 05 · Combinatorics

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

Source reports OPENAs of 2026-09-08Source & evidence ↓
Read original source ↗ 0Submit progressLean 4 ↓Discussion (0)

Problem statement

Let N(k,ℓ)N(k,\ell) be the minimal NN such that for any f:{1,…,N}→{−1,1}f:\{1,\ldots,N\}\to\{-1,1\} there must exist a kk-term arithmetic progression PP such that

∣∑n∈Pf(n)∣≥ℓ.\left\lvert \sum_{n\in P}f(n)\right\rvert\geq \ell.

Find good upper bounds for N(k,ℓ)N(k,\ell). Is it true that for any c>0c>0 there exists some C>1C>1 such that

N(k,ck)≤Ck?N(k,ck)\leq C^k?

What about

N(k,2)≤CkN(k,2)\leq C^k

or

N(k,k)≤Ck?N(k,\sqrt{k})\leq C^k?

Lean 4

Not yet attached

No Lean statement or proof is attached yet. Have a formalization? Share its source to add it to this problem.

Share a formalization →Verification guide ↗

Background

When ℓ=k\ell=k this is the van der Waerden number W(k)W(k) (see [138]). Spencer [Sp73] has proved that if k=2tmk=2^tm with mm odd then

N(k,1)=2t(k−1)+1.N(k,1)=2^t(k-1)+1.

Erdős and Graham write that 'no decent bound' is known even for N(k,2)N(k,2).

Erdős [Er63d] proved that, for every c>0c>0,

N(k,ck)>(1+αc)kN(k,ck)> (1+\alpha_c)^k

where αc→0\alpha_c\to 0 as c→0c\to 0 and αc→2−1\alpha_c\to \sqrt{2}-1 as c→1c\to 1.

Hunter in the comment section observes that the local lemma implies an improved bound of

N(k,ck)≫2kkO(1)∑i>1+c2k(ki),N(k,ck) \gg \frac{2^k}{k^{O(1)}\sum_{i>\frac{1+c}{2}k}\binom{k}{i}},

so that in particular as c→1c\to 1 we have, for all large kk, N(k,ck)≥(2−o(1))kN(k,ck)\geq (2-o(1))^k (where the o(1)o(1) term →0\to 0 as c→1c\to 1).

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 #176 link

Recorded progress

Add yours →

Nothing recorded yet. A pointer to a paper the entry is missing counts.

Machine attempts

0 recorded

Attempts run against the statement below and are recorded in full, including the failures. The point is to stop the next model — and the next person — from re-walking a road that is already known to end. Each attempt costs 25 credits.

No machine attempts recorded.

Sign in to run an attempt. New accounts start with enough credits for ten.

Discussion

0 notes

Notes are for clarifying the statement, flagging a missing reference, or recording why an approach fails. Claimed solutions belong in a submission, where they can be reviewed.

No notes yet.

Sign in to add a note.