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Artículo
3-color Schur numbers
(Elsevier, 2019)
Let k ≥ 3 be an integer, the Schur number Sk(3) is the least positive integer, such that for every 3-coloring of the integer interval [1, Sk(3)] there exists a monochromatic solution to the equation x1+ · · · + xk= xk+1, ...
Artículo
Exact values and lower bounds on the n-color weak Schur numbers for n = 2, 3
(Springer, 2023-08-23)
For integers k, n with k, n ≥ 1, the n-color weak Schur number WSk (n) is defined as the least integer N, such that for every n-coloring of the integer interval [1, N], there exists a monochromatic solution x1,..., xk , ...
Artículo
On the finiteness of some n-color Rado numbers
(Elsevier, 2017)
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 ...
Artículo
Exact value of 3 color weak Rado number
(Elsevier, 2016)
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 ...
Artículo
On the n-Color Weak Rado Numbers for the Equation x1 + x2 + ··· + xk + c = xk +1
(Taylor and Francis, 2019)
For integers k, n, c with k, n ≥ 1, and c ≥ 0, the n-color weak Rado number WRk (n, c) is defined as the least integer N, if it exists, such that for every n-coloring of the integer interval [1, N], there exists ...
Artículo
Weak Schur numbers and the search for G.W. Walker’s lost partitions
(Elsevier, 2012)
A set A of integers is weakly sum-free if it contains no three distinct elements x, y, z such that x + y = z. Given k ≥ 1, let WS(k) denote the largest integer n for which {1, . . . , n} admits a partition into k weakly ...
Artículo
On the n-Color Weak Rado Numbers for the Equation x1 + x2 + ··· + xk + c = xk +1
(American Mathematical Society, 2016)
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 it exists or ∞ otherwise, such that for every n-coloring of the set {1, 2,...,N}, there exists a monochromatic ...
Artículo
A general lower bound on the weak Schur number
(Elsevier, 2018)
For integers k, n with k, n ≥ 1, the n-color weak Schur number W Sk(n) is defined as the least integer N, such that for every n-coloring of the integer interval [1, N], there exists a monochromatic solution x1, . . . , ...
Artículo
Lower bounds and exact values of the 2-color off-diagonal generalized weak Schur numbers WS(2;k,k) (Brief Announcement)
(Elsevier, 2023)
In this study, we focus on the concept of the 2-color off-diagonal generalized weak Schur numbers, denoted as WS(2; k1, k2). These numbers are defined for integers ki ≥ 2, where i = 1, 2, as the smallest integer M, such ...
Artículo
Modular Schur numbers
(Electronic Journal of Combinatorics, 2013)
For any positive integers l and m, a set of integers is said to be (weakly) l-sum free modulo m if it contains no (pairwise distinct) elements x1, x2, . . . , xl , y satisfying the congruence x1 + . . . + xl ≡ y mod m. ...