Rado Numbers for the Equation ∑i=1m−1xi+c=xm, for Negative Values of c

Kosek, Wojciech; Schaal, Daniel

doi:10.1006/aama.2001.0762

Cited by 17 publications

(16 citation statements)

References 2 publications

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“…where k + m − 2 ≤ 2ℓ and the pair (k + m − 2, ℓ) is none of (3, 2), (4, 2), (5, 3), (10,5), (11,6), (12,7), (13,8), (14,9). Therefore, we have…”

Section: Fact 2 Nmentioning

confidence: 98%

“…Theorem 1.10. Let the pair (k+m−2, ℓ) be none of (3, 2), (4, 2), (5, 3), (10,5), (11,6), (12,7), (13,8), (14,9) and ℓ ≥ 2.…”

Section: Valuesmentioning

confidence: 99%

“…Harborth and Maasberg [6] studied the 2-color Rado numbers of the equation a(x + y) = bz. Kosek and Schaal [11] investigated the 2-color Rado numbers of the equation m−1 i=1 x i + c = x m for negative values of c. Hopkins and Schaal [7] studied the Rado numbers for m−1 i=1 a i x i = x m . Adhikari et al [1] and Boza et al [2] got the results for the k-color Rado number for the equation…”

Section: Introductionmentioning

confidence: 99%

“…For more details on the Rado and Schur numbers, we refer to the book [13] and papers [1,2,6,7,8,10,11,9,15,16,18,19,25].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Some multivariable Rado numbers

Ye¹,

Mao²,

He³

et al. 2022

Preprint

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The Rado number of an equation is a Ramsey-theoretic quantity associated to the equation. Let E be a linear equation. Denote by R r (E) the minimal integer, if it exists, such that any r-coloring of [1, R r (E)] must admit a monochromatic solution to E. In this paper, we give upper and lower bounds for the Rado number of m−2 i=1 x i + kx m−1 = ℓx m , and some exact values are also given. Furthermore, we derive some results for the cases that ℓ = m = 4 and m = 5, ℓ = k + i (1 ≤ i ≤ 5). As a generalization, the r-color Rado numbers for linear equations E 1 , E 2 , ..., E r is defined as the minimal integer, if it exists, such that any r-coloring of [1, R r (E 1 , E 2 , . . . , E r )] must admit a monochromatic solution to some E i , where 1 ≤ i ≤ r. A lower bound for R r (E 1 , E 2 , . . . , E r ) and the exact values of R 2 (x + y = z, ℓx = y) and R 2 (x + y = z, x + a = y) were given by Lovász Local Lemma.

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“…where k + m − 2 ≤ 2ℓ and the pair (k + m − 2, ℓ) is none of (3, 2), (4, 2), (5, 3), (10,5), (11,6), (12,7), (13,8), (14,9). Therefore, we have…”

Section: Fact 2 Nmentioning

confidence: 98%

“…Theorem 1.10. Let the pair (k+m−2, ℓ) be none of (3, 2), (4, 2), (5, 3), (10,5), (11,6), (12,7), (13,8), (14,9) and ℓ ≥ 2.…”

Section: Valuesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…For more details on the Rado and Schur numbers, we refer to the book [13] and papers [1,2,6,7,8,10,11,9,15,16,18,19,25].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Some multivariable Rado numbers

Ye¹,

Mao²,

He³

et al. 2022

Preprint

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show abstract

“…In recent years the exact Rado numbers for several families of equations and inequalities have been found, but almost entirely for 2-colorings [1,2,7,13]. Several other problems related to Schur number and Rado numbers have also been considered [6,8].…”

Section: Introductionmentioning

confidence: 98%

Multiplicity of monochromatic solutions to x+y<z

Kosek

Robertson

Sabo

et al. 2010

Journal of Combinatorial Theory, Series A

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The Two-Colour Rado Number for the Equation ax + by = (a + b)z

Gupta

Rangan²,

Tripathi

2015

Ann. Comb.

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For relatively prime positive integers a and b, let n = R(a, b) denote the least positive integer such that every 2-colouring of [1, n] admits a monochromatic solution to ax + by = (a + b)z with x, y, z distinct integers. It is known that R(a, b) ≤ 4(a + b) + 1. We show that R(a, b) = 4(a + b) + 1, except when (a, b) = (3, 4) or (a, b) = (1, 4k) for some k ≥ 1, and R(a, b) = 4(a + b) − 1 in these exceptional cases.

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Rado Numbers for the Equation ∑i=1m−1xi+c=xm, for Negative Values of c

Abstract: For every integer m ≥ 3 and every integer c, let r m c be the least integer, if it exists, such that for every 2-coloring of the set 1 2 r m c there exists a monochromatic solution to the equation

Cited by 17 publications

References 2 publications

Some multivariable Rado numbers

Some multivariable Rado numbers

Multiplicity of monochromatic solutions to x+y<z

The Two-Colour Rado Number for the Equation ax + by = (a + b)z

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