Exact value of 3 color weak Rado number

For integers k, n, c with k, n ≥ 1 and c ≥ 0, the n color weak Rado number W Rk(n, c) is defined as the least integer N, if it exists, such that for every n coloring of the set {1, 2, ..., N}, there exists a monochromatic solution in that set to the equation x1 + x2 + ... + xk + c = xk+1, such that xi = xj when i = j. If no such N exists, then W Rk(n, c) is defined as infinite. In this work, we consider the main issue regarding the 3 color weak Rado number for the equation x1 + x2 + c = x3 and the exact value of the W R2(3, c) = 13c + 22 is established

On the finiteness of some n-color Rado numbers

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

On monochromatic linear recurrence sequences

In this paper we prove some van der Waerden type theorems for linear recurrence sequences. Under the assumption $a_{i-1}\leq a_{i}a_{s-1}$ ($i=2,\ldots,s$), we extend the results of G. Nyul and B. Rauf \cite{nyul} for sequences satisfying $x_i=a_1x_{i-s}+\ldots+a_sx_{i-1}$ ($i\geq s+1$), where $a_{1},\ldots,a_{s}$ are positive integers. Moreover, we solve completely the same problem for sequences satisfying the binary recurrence relation $x_i=ax_{i-1}-bx_{i-2}$ ($i\geq 3$) and x_1<x_2, where $a,b$ are positive integers with $a\geq b+1$