Restricted generalized Schur numbers: exact values at levels three and four, and the threshold of Open Question 6.4

Abstract

For integers r ≥ 1, k ≥ 2 and ℓ ≥ 2, the restricted generalized Schur number Sᵣ(k;ℓ) is the least n such that every r-colouring of {1,…,n} admits a monochromatic solution of x₁+…+xₖ=xₖ₊₁ in which exactly ℓ+1 distinct integers occur. Gaiser introduced these numbers and proved that for each fixed ℓ ≥ 2 one has S₂(k;ℓ)=F(k,ℓ):=k²+[((ℓ+1)(ℓ-2))/(2)+2]k+ℓ(ℓ-2) for all sufficiently large k, without determining how large; his Open Question 6.4 asks for the least such threshold K. We compute twenty-one values of Sᵣ(k;ℓ) exactly, ten of them new: S₂(k;3) for 4 ≤ k ≤ 9 and S₂(k;4) for 5 ≤ k ≤ 8. Making the arithmetic of Gaiser's Section 5 explicit, we show that his own sufficient condition for the upper bound holds at level 3 exactly when k ≥ 10, and at level 4 exactly when k ≥ 20. Since our values cover every k from 3 to 10 at level 3, this answers Open Question 6.4 there: the least K is 3, the smallest value the definition permits. At level 4 the same reading leaves the eleven values 9 ≤ k ≤ 19 open. In the opposite direction we prove the identity 2F(k,k)+(k-1)(k-2)=k³+4k²-5k+2, whose right-hand side is twice a published lower bound for the weak Schur number S₂(k;k); consequently K>ℓ for every ℓ ≥ 5, and the closed form cannot begin at k=ℓ except at levels 3 and 4. All statements are machine-checked in Lean 4.

Open review

This founding-collection manuscript received AI review before publication. Independent human review is open. Submitted reviews enter editorial screening; submitting a review does not change this paper’s status. Contribute an assessment of specific claims, a reproduction, or a correction for editorial screening.

Archived files

1. Version 1 · current (opens in a new tab)

   Source snapshot 2026-08-30 15:34 UTC

   File fingerprint

   db1f5c0cfa8abdf76a8579f3817bb6ccf960b102aa70f71fbccb2fddee274ef4