THE FOLD / RESPAWN / SECOND WIND / THE VAN DER WAERDEN
THE VAN DER WAERDEN
order you cannot avoid
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Colour the numbers 1 to 8 red and blue however you like, and you can avoid ever having three evenly spaced numbers all the same colour. Add a single number — go to 9 — and it becomes impossible. Every one of the 512 colourings contains a monochromatic arithmetic progression. That threshold is W(3,2) = 9, and van der Waerden proved in 1927 that such a threshold exists for every number of colours and every length. The catch: the thresholds grow so violently that W(6,2) is still unknown — the best general bound, from Gowers, is a tower of exponentials.
LIT verified live by complete enumeration of every 2-colouring: n=3 → 6 of 8 avoid, n=4 → 10/16, n=5 → 14/32, n=6 → 20/64, n=7 → 16/128, n=8 → 6 of 256 still avoid, n=9 → 0 of 512, n=10 → 0 of 1024. The last n admitting an avoider is 8 and the first admitting none is 9, so W(3,2) = 9 exactly; a witness at n=8 is exhibited (11001100) and re-checked against all 12 three-term progressions (window.__vanderwaerden).
LIT verified live by complete enumeration of every 2-colouring: n=3 → 6 of 8 avoid, n=4 → 10/16, n=5 → 14/32, n=6 → 20/64, n=7 → 16/128, n=8 → 6 of 256 still avoid, n=9 → 0 of 512, n=10 → 0 of 1024. The last n admitting an avoider is 8 and the first admitting none is 9, so W(3,2) = 9 exactly; a witness at n=8 is exhibited (11001100) and re-checked against all 12 three-term progressions (window.__vanderwaerden).
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-gauntlet’s cousin, second-wind — the respawn: you survive the run at length 8 by the skin of your teeth, six ways out of 256, and at length 9 there is no run that survives at all. The margin does not shrink gradually; it hits zero. AVAN (AI) built the instrument: the full colouring enumerator, the progression detector, and the witness re-check.
Credit as content: B. L. van der Waerden (1927); Timothy Gowers (2001, the tower-of-exponentials bound); Michal Kouril (the computational values of W(6,2) still out of reach). The weave: David names the last survivable run; I enumerate every colouring and watch the survivors go to zero.
Credit as content: B. L. van der Waerden (1927); Timothy Gowers (2001, the tower-of-exponentials bound); Michal Kouril (the computational values of W(6,2) still out of reach). The weave: David names the last survivable run; I enumerate every colouring and watch the survivors go to zero.
3 ONE DIMENSION
Survivors by length — 6 at n=8, none at n=9.
4 TWO DIMENSIONS · INTERACTIVE
Try colourings at n=9; every one contains a progression.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the survivor count collapsing to zero.
AVAN’s addition (the inverse-companion): don’t try to build disorder — measure how long you can afford it. The inverse of ‘can I avoid the pattern?’ is ‘disorder has a budget, and Ramsey theory prices it’: complete structurelessness is not available at any size past the threshold, no matter how cleverly you colour. Magenta is the pattern you cannot refuse; green is the six colourings at n=8 that were the last free choices. Randomness is a finite resource.
LIT Verified live by complete enumeration of every 2-colouring: n=3→6/8 avoid, n=4→10/16, n=5→14/32, n=6→20/64, n=7→16/128, n=8→6 of 256 still avoid, n=9→0 of 512, n=10→0 of 1024. Last n admitting an avoider is 8, first admitting none is 9, so W(3,2) = 9 exactly; a witness at n=8 (11001100) is re-checked against all 12 three-term progressions (window.__vanderwaerden.ok).
FIG van der Waerden 1927; Gowers 2001 for the tower-of-exponentials bound; W(5,2)=178 known, W(6,2) unknown — cited, not computed. The AVAN inverse — measure how long you can AFFORD disorder: complete structurelessness is unavailable past the threshold no matter how cleverly you colour. Randomness is a finite resource.
FIG van der Waerden 1927; Gowers 2001 for the tower-of-exponentials bound; W(5,2)=178 known, W(6,2) unknown — cited, not computed. The AVAN inverse — measure how long you can AFFORD disorder: complete structurelessness is unavailable past the threshold no matter how cleverly you colour. Randomness is a finite resource.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SECOND WIND · David Lee Wise (ROOT0), with AVAN