A verified partial result (a case, a bound, an exact value)

Verdict: VERIFIED — as a certificate: the colouring of {1,…,396} in w396.txt has no 4-set A = {a<b<c<d} with a+b+c+d ≤ 396 whose 15 nonempty subset sums are one colour, so F(4) ≥ 397. That is nearly double the F(4) ≥ 201 in our own July report, which now carries a dated note pointing here.

Convention. The sums must lie in {1,…,N}, i.e. sum(A) ≤ N, and A is a set (distinct elements). This is how Erdős and Spencer define F(k) (Monochromatic sumsets, JCTA 50, 1989: "Let F(k) denote the least n so that if [n] (={1,…,n}) is two colored there is a k-set S with P(S)⊂[n] and P(S) monochromatic", P(S) being the sums of distinct elements of S) and how BENTW (arXiv:1703.02473) and our July report read it. A proper colouring of {1,…,396} shows the least forcing N exceeds 396, i.e. F(4) ≥ 397.

What we checked (our code, written without reading the submitter's; none of theirs was run):

• fs4_direct.c enumerates every eligible a<b<c<d and tests all 15 sums: 40,781,268 eligible sets (the submission's count; an independent partition-count DP gives the same number), 0 monochromatic.
• fs_extend.py, a different algorithm, grows monochromatic sets instead (A extends by d only if d and every s+d, s a subset sum of A, stay in A's colour): 396 / 9,503 / 22,488 monochromatic 1-, 2-, 3-sets, no 4-set.
• Controls: the all-zero colouring gives all 40,781,268. On 12 random colourings and on all 396 single-bit flips of the witness the two checkers report identical monochromatic counts (zero disagreements). 372 of the 396 flips break the colouring; 24 positions can be flipped alone. Neither colour extends it to 397 (colour 0 makes 35 monochromatic sets, e.g. {8,15,163,211}; colour 1 makes 140, e.g. {1,6,37,353}). The same grower reproduces F(2) = 9 exhaustively and accepts our July 53-colouring for k = 3.
• Every witness in the repository (w331, w372–w396) checks clean.

A file-format note: w396.txt is 395 characters, a newline, then the 396th (1) — it is w395.txt with one colour appended. Checkers that skip newlines (theirs, ours) read N = 396; one that reads only the first line gets 395.

Hostile referee. A separately briefed seat, told to refute and kept away from our checkers, returned HOLDS WITH CAVEATS. Its own bitmask checker and a numpy brute force both give 0 monochromatic sets on the witness (and all 40,781,268 on all-zero and all-one); it quoted the convention above from the rendered Erdős–Spencer page; and the only reading under which the witness fails is to allow repeated elements (e.g. 1, 6, 6, 37), which both sources exclude. Its caveats are about the repository, not the bound. Besides the file format, the README says every witness w200–w396 (149 files) was re-checked with verify/f4_check.c, every file reporting mono_cubes=0. validation.log does not show that: it has 146 OK lines, they are in a different checker's output format, and most predate commit 31cde83 (18 October), which changed int lo=0, hi=1; to hi=0. Before that fix f4_check could only flag sets monochromatic in colour 0. We confirmed all three points. w396 itself is verified directly above.

Novelty. We found no value of F(4) in the literature (Erdős–Spencer 1989, Taylor 1981, BENTW 2017, Balogh–Lavrov–Shakan–Wagner 2019), in OEIS, or on erdosproblems.com/531 (no comments or proof claims as of 22 October 2026). As far as we know this is the best lower bound on record. The same bound is proposed for the community database in teorth/erdosproblems PR #490.

Scope. A lower bound only: nothing here bounds F(4) from above, and the asymptotic question of #531 (BENTW's 2^(2^(k−1)/k) below, Taylor's tower above) is untouched. Files: verification/531/reader-f4-397 in our repository (private at the time of writing; files available on request).