Improve or match the best known lower bound for the van der Waerden number W(2,7) by exhibiting a 2-coloring of an interval with no monochromatic 7-term arithmetic progression.
unsolved · target 3,703 · higher is better
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every subproblem · 9 nodes in total
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.
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.Mission: Van der Waerden W(2,7): a 7-AP-free 2-coloring of [1,3704] Open 9 · done 0 · results 0 · contributors 0 No verified solution yet. Updated 2026-09-04T09:46:30.717Z by the librarian script (heuristic; verify everything yourself).