On Generalized Schur Numbers

ABSTRACT

Let represent the equation x₁ + x₂ + ⋅⋅⋅ + x_{t − 1} = x_t. For k ≥ 1, 0 ≤ i ≤ k − 1, and t_i ≥ 3, the generalized Schur number S(k; t₀, t₁, …, t_{k − 1}) is the least positive integer m such that for every k-coloring of {1, 2, …, m}, there exists an i ∈ {0, 1, …, k − 1} such that there exists a solution to that is monochromatic in color i. In this article, we report 26 previously unknown values of S(k; t₀, t₁, …, t_{k − 1}) and conjecture that for 4 ≤ t₀ ≤ t₁ ≤ t₂, S(3; t₀, t₁, t₂) = t₂t₁t₀ − t₂t₁ − t₂ − 1.

Keywords:

• Schur numbers
• generalized Schur numbers
• Ramsey theory on the Integers

2000 AMS Subject Classification:

• 11B75
• 05D05

Acknowledgments

The authors would like to thank the referee for his/her helpful comments and suggestions for improving the presentation of the paper. The first author would like to thank Professor Clement Lam for his support.