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

1/9≤ value ≤no upper bound trackedlower boundupper bound
10−5​0.0010.16 Octupdated on Oct 16, 2026Lower bound 1/100000, Sep 23, 2026, OpenAI, publishedLower bound 1/200, Oct 9, 2026, rohanarun, mergedLower bound 1/193, Oct 9, 2026, rohanarun, mergedLower bound 1/25, Oct 10, 2026, rohanarun, mergedLower bound 1/19, Oct 14, 2026, artuskg, mergedLower bound 1/16, Oct 14, 2026, DaysSky, mergedLower bound 1/9, Oct 14, 2026, DaysSky, merged1/9lower10−5​0.0010.127 Sept2 Oct7 Oct12 Octupdated on Oct 16, 2026Lower bound 1/100000, Sep 23, 2026, OpenAI, publishedLower bound 1/200, Oct 9, 2026, rohanarun, mergedLower bound 1/193, Oct 9, 2026, rohanarun, mergedLower bound 1/25, Oct 10, 2026, rohanarun, mergedLower bound 1/19, Oct 14, 2026, artuskg, mergedLower bound 1/16, Oct 14, 2026, DaysSky, mergedLower bound 1/9, Oct 14, 2026, DaysSky, merged1/9lower 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.
  1. lower bound ≈ 0.052632by artuskgPR #4
  2. lower bound 0.062500by DaysSkyPR #6
  3. lower bound ≈ 0.11111by DaysSkyPR #7
  4. Last 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 16, 2026]. DiscoverIA registry. https://discoveria.ai/en/frontieres/van-der-waerden/

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

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
DateExact valueNatureAuthorsSource
Oct 14, 20261/9 Current recordmergedDaysSkyPR #7
Oct 14, 20261/16 Current recordmergedDaysSkyPR #6
Oct 14, 20261/19 Current recordmergedartuskgPR #4
Oct 10, 20261/25 Current recordmergedrohanarunPR #2
Oct 9, 20261/193 Current recordmergedrohanarunPR #1
Oct 9, 20261/200 Current recordmergedrohanarungithub.com
Sep 23, 20261/100000 Current recordpublishedOpenAIgithub.com
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