What this page is
For years the community's reference tables for van der Waerden data were maintained by Tanbir Ahmed, whose computations produced many of the values below; his pages have since gone offline. This page keeps a maintained, verifiable successor for the five mixed families I work on: every known term as the OEIS records it, every new term with a certificate you can check yourself, and every claim labelled by the kind of evidence behind it.
- Lower bounds are certificates. Each new term links a witness colouring; checking it needs the definition and nothing else.
- Upper bounds are labelled honestly. All five new refutations have been reduced to formally checked proof objects: 23,851 per-cube DRAT proofs across the five families, each replayed to s VERIFIED by drat-trim, plus a checked proof for each family that its cube set is exhaustive. The tables label every upper bound by the evidence behind it rather than dressing a verdict as a proof.
The tables are generated from the OEIS and cross-checked against the b-files in MathRecords by a script that refuses to build the page if the two sources disagree — last regenerated 26 August 2026.
The five families
A217058
Van der Waerden numbers w(j+2; t0,t1,…,tj-1, 3, 4) with t0 = t1 = … = tj-1 = 2.
| Term | Value | Provenance |
|---|---|---|
| a(0) | 18 | previously known — as recorded by the OEIS |
| a(1) | 21 | previously known — as recorded by the OEIS |
| a(2) | 25 | previously known — as recorded by the OEIS |
| a(3) | 29 | previously known — as recorded by the OEIS |
| a(4) | 33 | previously known — as recorded by the OEIS |
| a(5) | 36 | previously known — as recorded by the OEIS |
| a(6) | 40 | previously known — as recorded by the OEIS |
| a(7) | 42 | previously known — as recorded by the OEIS |
| a(8) | 45 | previously known — as recorded by the OEIS |
| a(9) | 48 | previously known — as recorded by the OEIS |
| a(10) | 52 | previously known — as recorded by the OEIS |
| a(11) | 55 | previously known — as recorded by the OEIS |
| a(12) | 57 | new — published by the OEIS, credited 30 July 2026; witness colouring for the lower bound; upper bound reduced to checked DRAT proof objects |
A217005
Van der Waerden numbers w(j+2; t0,t1,…,tj-1, 3, 3) with t0 = t1 = … = tj-1 = 2.
| Term | Value | Provenance |
|---|---|---|
| a(0) | 9 | previously known — as recorded by the OEIS |
| a(1) | 14 | previously known — as recorded by the OEIS |
| a(2) | 17 | previously known — as recorded by the OEIS |
| a(3) | 20 | previously known — as recorded by the OEIS |
| a(4) | 21 | previously known — as recorded by the OEIS |
| a(5) | 24 | previously known — as recorded by the OEIS |
| a(6) | 25 | previously known — as recorded by the OEIS |
| a(7) | 28 | previously known — as recorded by the OEIS |
| a(8) | 31 | previously known — as recorded by the OEIS |
| a(9) | 33 | previously known — as recorded by the OEIS |
| a(10) | 35 | previously known — as recorded by the OEIS |
| a(11) | 37 | previously known — as recorded by the OEIS |
| a(12) | 39 | previously known — as recorded by the OEIS |
| a(13) | 42 | previously known — as recorded by the OEIS |
| a(14) | 44 | previously known — as recorded by the OEIS |
| a(15) | 46 | previously known — as recorded by the OEIS |
| a(16) | 48 | previously known — as recorded by the OEIS |
| a(17) | 50 | previously known — as recorded by the OEIS |
| a(18) | 51 | previously known — as recorded by the OEIS |
| a(19) | 52 | new — published by the OEIS, credited 7 August 2026; witness colouring for the lower bound; upper bound reduced to checked DRAT proof objects |
A217007
Van der Waerden numbers w(j+2; t0,t1,…,tj-1, 4, 4) with t0 = t1 = … = tj-1 = 2.
| Term | Value | Provenance |
|---|---|---|
| a(0) | 35 | previously known — as recorded by the OEIS |
| a(1) | 40 | previously known — as recorded by the OEIS |
| a(2) | 53 | previously known — as recorded by the OEIS |
| a(3) | 54 | previously known — as recorded by the OEIS |
| a(4) | 56 | previously known — as recorded by the OEIS |
| a(5) | 66 | previously known — as recorded by the OEIS |
| a(6) | 67 | previously known — as recorded by the OEIS |
| a(7) | 68 | new — published by the OEIS, credited 7 August 2026; witness colouring for the lower bound; upper bound reduced to checked DRAT proof objects |
A217236
Van der Waerden numbers w(j+2; t0,t1,…,tj-1, 4, 5) with t0 = t1 = … = tj-1 = 2.
| Term | Value | Provenance |
|---|---|---|
| a(0) | 55 | previously known — as recorded by the OEIS |
| a(1) | 71 | previously known — as recorded by the OEIS |
| a(2) | 75 | previously known — as recorded by the OEIS |
| a(3) | 79 | previously known — as recorded by the OEIS |
| a(4) | 84 | new — published by the OEIS, credited 7 August 2026; witness colouring for the lower bound; upper bound reduced to checked DRAT proof objects |
A217059
Van der Waerden numbers w(j+2; t0,t1,…,tj-1, 3, 5) with t0 = t1 = … = tj-1 = 2.
| Term | Value | Provenance |
|---|---|---|
| a(0) | 22 | previously known — as recorded by the OEIS |
| a(1) | 32 | previously known — as recorded by the OEIS |
| a(2) | 43 | previously known — as recorded by the OEIS |
| a(3) | 44 | previously known — as recorded by the OEIS |
| a(4) | 50 | previously known — as recorded by the OEIS |
| a(5) | 55 | previously known — as recorded by the OEIS |
| a(6) | 61 | previously known — as recorded by the OEIS |
| a(7) | 65 | previously known — as recorded by the OEIS |
| a(8) | 70 | previously known — as recorded by the OEIS |
| a(9) | 74 | new — published by the OEIS, credited 13 August 2026; witness colouring for the lower bound; upper bound reduced to checked DRAT proof objects |
Verify it yourself
No claim on this page asks for trust. The witness colourings check in milliseconds with verify_certificate.py; the whole repository re-derives every published value from scratch under verify_all.py, in CI; and the DRAT machinery — including the measured cost of each proof and why colour symmetry had to come out of the cube set before the equal-target families could be certified — is documented in DRAT.md.
References
- T. Ahmed, Some new van der Waerden numbers and some van der Waerden-type numbers, Integers 9 (2009); and Some More Van der Waerden Numbers, J. Integer Seq. 16 (2013) — the prior work these families stood on for over a decade.
- T. Ahmed, On the van der Waerden numbers w(2;3,t), arXiv:1102.5433.
- The OEIS entries linked in each table, which carry the full submission history of every term.
- Five new mixed van der Waerden numbers — the write-up of the 2026 terms, with the encoding, the certificates and an explicit account of which claims rest on which kind of evidence: MathRecords/paper.