A note on disjunctive Rado numbers

Kosek, Wojciech; Schaal, Daniel

doi:10.1016/s0196-8858(03)00020-4

Cited by 3 publications

(11 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…, 7 ∈ R. The solutions (2, 3, 1, 10), (4, 5, 1, 11), (13,4,1,13), (24,2,2,24), and (82, 1, 7, 82) show that 10, 11, 13, 24, 82 ∈ B. Then we have 8, 93, 94 ∈ R by considering (24,24,8,82), (11,11,10,93), and (13,13,10,94). Therefore, (3,93,8,94) is a red solution, a contradiction.…”

Section: Results For Four Variable Rado Numbersmentioning

confidence: 96%

“…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%

“…Suppose that there exists a 2-coloring of [1, B − 1] that admit a monochromatic solution to We now give the proof of Theorem 1.10 by the following four lemmas. 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: Fact 2 Nmentioning

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 4 more Smart Citations

Some multivariable Rado numbers

Ye¹,

Mao²,

He³

et al. 2022

Preprint

View full text Add to dashboard Cite

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.

show abstract

Section: Results For Four Variable Rado Numbersmentioning

confidence: 96%

“…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: Fact 2 Nmentioning

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

View full text Add to dashboard Cite

show abstract

New Proofs for the Disjunctive Rado Number of the Equations $$x_1-x_2=a$$ and $$x_1-x_2=b$$

Dileep

Moondra

Tripathi

2022

Graphs and Combinatorics

View full text Add to dashboard Cite

On Disjunctive Rado Numbers for Some Sets of Equations

Dileep,

Moondra,

Tripathi

2024

Electron. J. Combin.

View full text Add to dashboard Cite

Given a set of linear equations $S$ with positive integral parameters $a_1,\ldots,a_k$, $k \ge 2$, the disjunctive Rado number for the set $S$ is the least positive integer $R=R_d(S)$, if it exists, such that every $2$-coloring $\chi$ of the integers in $[1, R]$ admits a monochromatic solution to at least one equation in $S$. We give conditions for the existence of $R_d(S)$, and also give general upper and lower bounds on $R_d(S)$, when $S$ is a set of additive equations $\{y=x+a_1,\ldots,y=x+a_k\}$. We also determine $R_d(S)$ when $\max a_i$ is large enough, or when $a_1,\ldots,a_k$ form an arithmetic or geometric progression. We also give conditions for the existence of $R_d(S)$ when $S$ is a set of multiplicative equations $\{y=a_1 x,\ldots,y=a_k x\}$. Further, we give a general search-based algorithm to determine $R_d(S)$ when $S$ is a system of equations in two variables, given an upper bound on $R_d(S)$ and an algorithm to determine solutions to $S$. This algorithm runs in time $O(ka_k \log a_k)$ for the case of additive equations, which is exponentially better than the brute-force algorithm for the problem.

show abstract

A note on disjunctive Rado numbers

Cited by 3 publications

References 8 publications

Some multivariable Rado numbers

Some multivariable Rado numbers

New Proofs for the Disjunctive Rado Number of the Equations $$x_1-x_2=a$$ and $$x_1-x_2=b$$

On Disjunctive Rado Numbers for Some Sets of Equations

Contact Info

Product

Resources

About