Article
On the finiteness of some n-color Rado numbers
Author/s | Adhikari, S. D.
Boza Prieto, Luis Eliahou, Shalom Marín Sánchez, Juan Manuel Revuelta Marchena, María Pastora Sanz Domínguez, María Isabel |
Department | Universidad de Sevilla. Departamento de Matemática Aplicada I (ETSII) |
Publication Date | 2017 |
Deposit Date | 2022-07-29 |
Published in |
|
Abstract | For integers k, n, c with k, n ≥ 1, the n-color Rado number Rk(n, c) is defined to be
the least integer N if any, or infinity otherwise, such that for every n-coloring of the set
{1, 2, . . . , N}, there exists a ... For integers k, n, c with k, n ≥ 1, the n-color Rado number Rk(n, c) is defined to be the least integer N if any, or infinity otherwise, such that for every n-coloring of the set {1, 2, . . . , N}, there exists a monochromatic solution in that set to the linear equation x1 + x2 + · · · + xk + c = xk+1. A recent conjecture of ours states that Rk(n, c) should be finite if and only if every divisor d ≤ n of k−1 also divides c. In this paper, we complete the verification of this conjecture for all k ≤ 7. As a key tool, we first prove a general result concerning the degree of regularity over subsets of Z of some linear Diophantine equations. |
Citation | Adhikari, S.D., Boza Prieto, L., Eliahou, S., Marín Sánchez, J.M., Revuelta Marchena, M.P. y Sanz Domínguez, M.I. (2017). On the finiteness of some n-color Rado numbers. Discrete Mathematics, 340 (2), 39-. |
Files | Size | Format | View | Description |
---|---|---|---|---|
1-s2.0-S0012365X1630231X-main.pdf | 510.6Kb | [PDF] | View/ | |