bibkey: gaiser2026restrictedschur authors: Collier Gaiser year: 2026 title: “Restricted generalized Schur numbers” doi: 10.48550/arXiv.2608.08789 url: https://arxiv.org/abs/2608.08789v1 claim: “Introduces the restricted generalized Schur numbers S_r(k; l), determines S_2(k; l) for large k, proves S_3(k; 2) ≥ k^3 + 3k^2 + k − 1 (Proposition 6.1) and asks in Open Question 6.2 whether equality holds for all large k.” strata_touched:

• D5/S3/Combinatorics/RestrictedSchur/RestrictedSchur license: citation-only triage: anchor

Gaiser, restricted generalized Schur numbers

For k ≥ 2 the paper studies S_r(k; l), the least n such that every r-colouring of {1, …, n} has a monochromatic solution of x_1 + ⋯ + x_k = x_{k+1} with exactly l + 1 distinct integers. Theorem 1.1 determines S_2(k; l) for fixed l ≥ 2 and large k, in particular S_2(k; 2) = k^2 + 2k for k ≥ 3. Proposition 6.1 gives a block 3-colouring showing S_3(k; 2) ≥ k^3 + 3k^2 + k − 1, and Open Question 6.2 asks whether S_3(k; 2) = k^3 + 3k^2 + k − 1 for all large enough k.

The module D5/S3/Combinatorics/RestrictedSchur/RestrictedSchur answers Open Question 6.2 negatively.

Verified locator

DOI: 10.48550/arXiv.2608.08789

URL: https://arxiv.org/abs/2608.08789v1

• Locator: Section 1 (definition of S_r(k; l) and c(S)); Section 6, Proposition 6.1 and Open Question 6.2.