Monochromatic Arithmetic Progressions With Large Differences
Tom C. Brown,Bruce Landman +1 more
TL;DR: In this article, a generalisation of van der Waerden numbers w k r is considered and upper and lower bounds for w f 3 2 and w f k r 2 are given.
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Abstract: A generalisation of the van der Waerden numbers w k r is considered. For a function f : define w k f r to be the least positive integer (if it exists) such that for every r-coloring of 1 w f k r there is a monochromatic arithmetic progression a id : 0 i k 1 such that d f a . Upper and lower bounds are given for w f 3 2 . For k 3 or r 2, particular functions f are given such that w f k r does not exist. More results are obtained for the case in which f is a constant function.
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Citations
On generalized van der Waerden triples
Bruce Landman,Aaron Robertson +1 more
TL;DR: It is shown that for a large class of pairs (a,b), N(a, b;r) does not exist for r sufficiently large, and upper and lower bounds for it are provided.
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On the degree of regularity of generalized van der Waerden triples
TL;DR: The degree of regularity of the family of all ( a, b ) -triples, denoted dor ( a , b ) , is the maximum integer r such that every r-coloring of N admits a monochromatic ( a), b -triple.
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On the degree of regularity of generalized van der Waerden triples
TL;DR: In this paper, it was shown that dor(a,b) < √ √ 1,1/ √ (1,1) for all $(a, b)
eq (1 1/1/1) and also disproved the conjecture that the degree of regularity of the family of all (1 + b)-triples is at most (1 − 1,2, √ 2/1).
1
Some multivariable Rado numbers
Gang Yang,Yaping Mao,Chang-Xiang He,Zhao Wang +3 more
- 05 Mar 2022
TL;DR: In this paper , upper and lower bounds for the Rado number of ∑m−2 i = 1 xi + kxm−1 = lxm were given.
Monochromatic structures in colorings of the positive integers and the finite subsets of the positive integers
Tom C. Brown
- 01 Jan 2003
TL;DR: In this article, the authors discuss van der Waerden's theorem on arithmetic progressions and an extension using Ramsey's theorem, and the canonical versions of Theorem 6 and Theorem 7.
References
Polynomial extensions of van der Waerden’s and Szemerédi’s theorems
TL;DR: An extension of the van der Waerden and Szemeredi the- orems for commuting operators whose exponents are polynomials is proved in this paper, where the authors obtain the following result: for any vector u ∈ Zl such that u+ pi(n)vi ∈ S for each i ≤ k
On the Set of Common Differences in van der Waerden's Theorem on Arithmetic Progressions
TL;DR: The notion of large sets was introduced in this article, where the family of all arithmetic progressions, AP, is replaced by a subfamily of AP, AP-AP, which is called r-large.
A remark concerning arithmetic progressions
TL;DR: F(d) ⩽ (1 + e) log2 d is proved, which proves that if the authors split the integers into two classes at least one class contains, for infinitely many values of d, an arithmetic progression of difference d and length F(d).
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On van der Waerden's theorem and the theorem of Paris and Harrington
TL;DR: A 2-coloring of the non-negative integers and a function h are given such that if P is any monochromatic arithmetic progression with first term a and common difference d then ‖ P ‖ ⩽ h ( a ) and ‖ H ‖ h ( d) are given.
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On some generalizations of the van der Waerden number w (3)
TL;DR: Upper and lower bounds are found for the more general function f c ( b ) which, for c =1, improve the aforementioned upper bound to ⌈9 b /4⌉+9, and give the same lower bound of 2 b +10.
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