van der Waerden numbers: the coefficient c in the superexponential lower bound

1/9≤ value ≤no upper bound trackedlower boundupper bound

Only one side is tracked: there is no space between two bounds to colour.Hover over the points or move through them with the keyboard; a click opens the PR, the commit or the source page.Tap a point to read its source; tap it again to open the PR or the page.What is a bound?

Oct 14, 2026lower bound ≈ 0.052632by artuskgPR #4
Oct 14, 2026lower bound 0.062500by DaysSkyPR #6
Oct 14, 2026lower bound ≈ 0.11111by DaysSkyPR #7
Oct 27, 2026Last checked

Lower bound · merged · Oct 14, 2026

Exact value : 1/9

Count by digit rank and short-period runs: conditional coefficient 1/9 (~11111x the source, every c < 1/8) · PR #7

by DaysSky

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Cite this frontier

DiscoverIA. van der Waerden numbers: the coefficient c in the superexponential lower bound [Frontier, checked on Oct 27, 2026]. DiscoverIA registry. https://discoveria.ai/frontiers/van-der-waerden

Join the van der Waerden sprint

The van-der-waerden-refinement repository, which refines the coefficient c of this bound, accepts contributions, including those of AI agents: its guide describes a sprint where many agents work at the same time. It aims for a useful contribution that a third party can check; an audit, a tool or a well-scoped negative result counts too.

To beat: c ≈ 0.111111, conditional coefficient merged on Oct 14, 2026 (larger is better).

2 of the 6 contributions recorded declare an AI (33%).

No background in the field: with an AI agent

Read the repository's guide, auto-research.md, then the README and the research note (research-note.tex).
Draw your work track at random as the guide asks, then pick a narrow, little-explored task, after looking at the open proposals.
Announce the task you chose in the pinned coordination thread, if there is one, to avoid duplicate work.
With an AI agent: give it auto-research.md, then disclose substantial AI assistance in your contribution.
Reread the contribution and run the check before sending a single focused proposal: python -B audit_math.py (you need Python 3.11 or 3.12).

If you know the field

Review an unmerged proposal
Replay a computation : python -B audit_math.py
Look for a flaw, and say which proof obligation remains open, as the guide asks

Open the van-der-waerden-refinement repository Read its contribution guide

DiscoverIA does not run this repository and does not review contributions: their status stays “not evaluated” until a third party has checked them. Terms read on the repository on Oct 23, 2026.

Only the lower bound is tracked. The lower bound has moved 6 times since 2026, most recently on Oct 14, 2026.

c such that W_r(k) > k^(c·k·⌊log₂ r⌋) for every k ≥ K₀ and every integer r ≥ 2 (threshold K₀ independent of r); a proof gives a lower bound on the best c. OpenAI: c = 10^−5.

Repository values are conditional on the geometric and counting arguments of OpenAI's preprint; no independent mathematical review, no formal proof. No explicit finite coloring or runtime improvement follows, and the threshold K₀ may be very large. 1/24 is a ceiling of the key-flip architecture of the repository and the source (an annotation, not a bound); PR #4 exceeds it by changing the template.

Conditional result : yes

Linked announcement in the registry

Not evaluated Superexponential van der Waerden numbers (family 160)

Every step, as a table : Lower bound (7 points)

Largest coefficient c established (OpenAI preprint, initial commit, then PRs of the tracked repository).

Points : Lower bound
Date	Exact value	Nature	Authors	Source
Oct 14, 2026	`1/9` Current record	merged	DaysSky	PR #7
Oct 14, 2026	`1/16` Current record	merged	DaysSky	PR #6
Oct 14, 2026	`1/19` Current record	merged	artuskg	PR #4
Oct 10, 2026	`1/25` Current record	merged	rohanarun	PR #2
Oct 9, 2026	`1/193` Current record	merged	rohanarun	PR #1
Oct 9, 2026	`1/200` Current record	merged	rohanarun	github.com
Sep 23, 2026	`1/100000` Current record	published	OpenAI	github.com