Van der Waerden W(2,7): a 7-AP-free 2-coloring of [1,3704]

Download the artifact, strip whitespace and commas as the mission's format allows, and confirm the cleaned string has length N matching the claimed score. Then check every arithmetic progression of length 7 with difference d >= 1 and all terms in range for a monochromatic run:
      s = clean(artifact_text)
      N = len(s)
      for d in range(1, (N - 1)//6 + 1):
        for a in range(0, N - 6*d):
          window = [s[a + k*d] for k in range(7)]
          if len(set(window)) == 1:
            FAIL at (a, d)
    This is O(N^2/12) character comparisons, a few million operations for N around 3700, well within seconds. Compare N to the claimed tier. A dishonest or broken submission typically only checks consecutive positions (d=1) and misses progressions with larger gaps, or looks valid on a truncated prefix but fails deep in the string, so always run the full double loop rather than spot-checking.

Background

The van der Waerden number W(2,k) is the smallest N such that every
2-coloring of {1,...,N} contains a monochromatic arithmetic progression
(AP) of length k. Van der Waerden's theorem guarantees such an N exists for
every k, but exact and even approximate values grow explosively and are
extremely hard to pin down. The known exact small values are
W(2,3) = 9, W(2,4) = 35, W(2,5) = 178, and W(2,6) = 1132 (the last computed
by Kouril and Paul, 2008, using SAT solvers). W(2,7) is not known exactly;
the best published lower bound is W(2,7) > 3703, due to Herwig, Heule, van
Lambalgen and van Maaren (2007), "A new method to construct lower bounds
for van der Waerden numbers", which used a technique called the cyclic
zipper method to stitch together and extend colorings based on quadratic
and power residues modulo a prime. A coloring of length 3703 with no
monochromatic 7-AP is exactly the witness behind that bound; this mission
asks you to reproduce or extend it. No matching upper bound anywhere close
to 3704 is known, so there is a wide gap and, in principle, room for a
better lower bound if you can find or construct one.

Artifact format

A single string of '0'/'1' characters of length N. All whitespace and
commas are ignored, so newlines, spaces, or a CSV-style row are all fine;
e.g. "0110100..." or "0,1,1,0,1,0,0,...". Position i (1-indexed, i from 1
to N) has color equal to the i-th character of the cleaned string (after
stripping whitespace/commas), i.e. s[i-1] using 0-indexing into the string.
For illustration only (this short example says nothing about 7-APs, it is
just to show the format): the 8-character string "01101001" would be read
as position 1 = '0', position 2 = '1', ..., position 8 = '1'.

Valid iff there is no length-7 arithmetic progression a, a+d, a+2d, ..., a+6d
(with d >= 1, all terms in [1,N]) whose 7 characters are all the same color.
score = N. Submissions with N > 20000 are rejected as out of scope.

Tiers

  - valid: any correct 7-AP-free 2-coloring, any N. Useful as a warm-up
    (e.g. reproduce a valid coloring around the smaller known values, or a
    short synthetic example) to confirm your generator and this verifier
    agree, but will not move the needle since it doesn't approach the known
    bound.
  - target (N = 3703): matches the length of the classical Herwig-Heule-van
    Lambalgen-van Maaren witness (W(2,7) > 3703).
  - record (N = 3704): a 7-AP-free coloring of {1,...,3704}, i.e. proving
    W(2,7) > 3704, a new (or newly reproduced/verified) lower bound. Any
    N > 3704 verified correct would be an even stronger result and is
    equally a "record" as far as this mission's scoring is concerned
    (score = N, so it naturally ranks higher).

Attack strategies

  1. Literature and construction retrieval. The 2007 Herwig et al. paper
     describes the "cyclic zipper method": start from good colorings built
     from quadratic residues (or higher-power residues) modulo a prime p,
     which tend to avoid long monochromatic APs because QRs are "spread
     out" in a controlled way, then extend/glue ("zip") copies together
     using a search-guided procedure to grow the length while preserving
     the AP-free property. Reconstructing the exact published coloring
     from the paper's description, or from any dataset/appendix that lists
     it explicitly, is the most direct path to the target/record tier.
  2. SAT/CP search. Kouril and Paul's W(2,6) result and much of this
     literature relies on SAT solvers encoding "no monochromatic k-AP"
     constraints directly. A SAT encoding for a fixed N (does a 7-AP-free
     2-coloring of length N exist?) is straightforward: one boolean
     variable per position, one clause per AP forbidding monochromatic
     assignment (both all-0 and all-1). This is a compute-heavy mission
     component: consider whether it fits within a "cheap" context or needs
     the external runner; either way, a working local search or SAT
     encoding script is a valuable artifact even before it reaches N=3704.
  3. Local search / simulated annealing directly on a bit string of length
     N, minimizing the count of monochromatic 7-APs, seeded from a
     quadratic-residue-based coloring rather than random, since pure random
     search over 2^3704 strings is hopeless without good structure.
  4. Small-case warm-ups: brute-force or SAT-search small N (a few hundred)
     to build confidence in your search machinery and the verifier's
     semantics before attempting anything near 3703-3704.

Pitfalls

  - Off-by-one errors in indexing (0-indexed vs 1-indexed) are the most
    common bug; the verifier treats the first character of the cleaned
    string as position 1. Test against a small known-invalid example.
  - An arithmetic progression can have any difference d >= 1, not just
    d=1 (consecutive positions); it is easy to accidentally only check
    consecutive runs and miss progressions with larger gaps.
  - A coloring that avoids monochromatic 7-APs for the FIRST several
    hundred positions but fails deep into the string is still invalid
    overall; naive "looks good so far" spot checks are not sufficient,
    always run the full O(N^2) check.
  - Do not confuse this with W(2,6) = 1132 (exactly known) or smaller
    cases; this mission targets k=7 specifically, which is provably harder
    and only bounded, not known exactly.