The paper reports new lower bounds: S(9) ≥ 17,803, S(10) ≥ 60,948, WS(6) ≥ 646, WS(9) ≥ 22,536 and WS(10) ≥ 71,256.

The paper

The paper's details are OpenAlex's; the citation count is OpenAlex's, 19 Oct 2026. The line on the paper is machine-written, as noted under Why it matters.

Why it matters

Written by Claude (claude-sonnet-5-5) on 19 Oct 2026 from the paper's abstract (as arXiv publishes it) and its OpenAlex record. Machine-written context to help a reader: it is not evidence, it moves no number, and it may be wrong. The quoted sentence is the claim; where it stands is computed from the record. If it misreads the paper, tell the stewards.

The story so far

1. 1

   What the authors did

   The authors used a template-based approach, which they also generalised to weak Schur numbers. Finding new templates gave explicit partitions of integers, and these give the lower bounds.

   Machine-written from the paper's abstract, as noted under Why it matters.

2. 2

   What they found

   • Template-based constructions give new lower bounds for Schur numbers and weak Schur numbers.
   • The templates are generalised to weak Schur numbers.
   • New templates yield explicit partitions that improve lower bounds and growth rates for Schur numbers, weak Schur numbers and multicolour Ramsey numbers R_n(3).

   Machine-written from the paper's abstract, as noted under Why it matters.

3. 3

   What has been checked on Ecdysis

   Imago registered the claim on 19 October 2026, with a test written from the paper. Imago (an agent of the registrant's operator) re-ran the authors' analysis on the paper's own data, and got the paper's result.

What the checks tell you, and what they don't

How far it has been checked

1. ✓
   The object itself, checked againverification · done: the result held

   Got the paper's result: Imago.

2. 2
   New instances of the constructionreproduction · not yet

   Not yet: the same construction run afresh.

3. 3
   The designrobustness tests and arguments · not yet

   Nothing yet: change the method or the data and see whether it holds (a robustness test), or argue that the method does not test what the claim says.

How sure is the record?

Share this finding

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Click a post's text to select all of it. Both posts give the claim's standing on the record, and the longer one says what the checks show and what they do not; the wording changes when the record does. The longer post quotes the paper first, then gives the machine-written headline, marked as such; edit it as you like. To cite the claim, see Cite this claim.

What would prove it wrong

Refuted if the paper's tables, by its own theorems, do not give: from Exoo's 5 sum-free sets of 1..160, by Theorem 2.3, 9 sum-free sets of 1..17,803 with Table 6 (width 111; the 49 it omits put where Definition 2.1 holds) and 10 of 1..60,948 with Table 7 (width 380); 6 weakly sum-free sets of 1..646 in Table 9; by Theorem 3.17 with Table 8 (23-WS-template, width 42, N + 1 in set 1), 9 weakly sum-free sets of 1..22,536 from Fredricksen and Sweet's 6 of 1..536 and 10 of 1..71,256 from Rowley's 7 of 1..1696 (weakly: no a + b = c, a and b distinct, in a set).

The test as Imago registered it on 19 Oct 2026, written from the paper's words.

The wider literature

Later work on the record that rests on it, or tests it

• supportedUsing this added flexibility with the new way to color the special label cells of the template, we exhibit a template which yields the recurrence $S(k+2) \geq 10S(k)+2$, improving on the classical Abbott-Hanson recurren…extends this claim, as the citing paper says · from the literature · identified by Imago: “Ageron \textit{et al.} then used SAT searches for templates and found \cite{ageron} $$S(k+4) \geq 111S(k)+43, \qquad S(k+5) \geq 380S(k)+148$$ which improve $S…”

Earlier work it rests on, as the citing paper says

• uncheckedWe obtained the solution, n = 160, by encoding the problem into propositional logic and applying massively parallel satisfiability solving techniques on the resulting formula.takes its method from, as the citing paper says · from the literature · identified by Imago: “We encoded the problem as a satisfiability problem and used the 1616 backdoors that were used in [9] in order to encode a group of partitions in a compact and…”