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Satisfiability and computing van der Waerden numbers
, 2004
"... In this paper we bring together the areas of combinatorics and propositional satisfiability. Many combinatorial theorems establish, often constructively, the existence of positive integer functions, without actually providing their closed algebraic form or tight lower and upper bounds. The area o ..."
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Cited by 8 (3 self)
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of Ramsey theory is especially rich in such results. Using the problem of computing van der Waerden numbers as an example, we show that these problems can be represented by parameterized propositional theories in such a way that decisions concerning their satisfiability determine the numbers (function
Some new exact van der Waerden numbers
 Integers: Elec. J. Combinatorial Number Theory
"... For positive integers r, k0,k1,..., kr−1, the van der Waerden number w(k0,k1,..., kr−1) is the least positive integer n such that whenever {1, 2,...,n} is partitioned into r sets S0,S1,..., Sr−1, there is some i so that Si contains a kiterm arithmetic progression. We find several new exact values o ..."
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Cited by 13 (3 self)
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For positive integers r, k0,k1,..., kr−1, the van der Waerden number w(k0,k1,..., kr−1) is the least positive integer n such that whenever {1, 2,...,n} is partitioned into r sets S0,S1,..., Sr−1, there is some i so that Si contains a kiterm arithmetic progression. We find several new exact values
Bounds on some van der Waerden numbers
 Journal of Combinatorial Theory, Series A
"... Abstract For positive integers s and k 1 , k 2 , . . . , k s , the van der Waerden number w(k 1 , k 2 , . . . , k s ; s) is the minimum integer n such that for every scoloring of set {1, 2, . . . , n}, with colors 1, 2, . . . , s, there is a k i term arithmetic progression of color i for some i. ..."
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Cited by 7 (2 self)
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Abstract For positive integers s and k 1 , k 2 , . . . , k s , the van der Waerden number w(k 1 , k 2 , . . . , k s ; s) is the minimum integer n such that for every scoloring of set {1, 2, . . . , n}, with colors 1, 2, . . . , s, there is a k i term arithmetic progression of color i for some i
Article 13.4.4 Some More Van der Waerden Numbers
"... The van der Waerden number w(k;t0,t1,...,tk−1) is the smallest positive integer n suchthateverykcoloringofthesequence1,2,...,nyieldsamonochromaticarithmetic progression of length ti for some color i ∈ {0,1,...,k −1}. In this paper, we propose a problemspecific backtracking algorithm for computing ..."
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The van der Waerden number w(k;t0,t1,...,tk−1) is the smallest positive integer n suchthateverykcoloringofthesequence1,2,...,nyieldsamonochromaticarithmetic progression of length ti for some color i ∈ {0,1,...,k −1}. In this paper, we propose a problemspecific backtracking algorithm for computing
The 2color relative linear Van der Waerden numbers ✩
, 2007
"... We define the rcolor relative linear van der Waerden numbers for a positive integer r as generalizations of the polynomial van der Waerden numbers of linear polynomials. Especially we express a sharp upper bound of the 2color relative linear van der Waerden number Rf2(u1,u2,...,um: s1,s2,...,sk) i ..."
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We define the rcolor relative linear van der Waerden numbers for a positive integer r as generalizations of the polynomial van der Waerden numbers of linear polynomials. Especially we express a sharp upper bound of the 2color relative linear van der Waerden number Rf2(u1,u2,...,um: s1,s2,...,sk
Lower Bounds on van der Waerden Numbers: Randomized and DeterministicConstructive
"... The van der Waerden number W (k, 2) is the smallest integer n such that every 2coloring of 1 to n has a monochromatic arithmetic progression of length k. The existence of such an n for any k is due to van der Waerden but known upper bounds on W (k, 2) are enormous. Much effort was put into developi ..."
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Cited by 3 (0 self)
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The van der Waerden number W (k, 2) is the smallest integer n such that every 2coloring of 1 to n has a monochromatic arithmetic progression of length k. The existence of such an n for any k is due to van der Waerden but known upper bounds on W (k, 2) are enormous. Much effort was put
An Application of Lovikz ' Local LemmaA New Lower Bound for the van der Waerden Number
"... The van der Waerden number W(n) is the smallest integer so that if we divide the integers {1,2,..., W(n)} into two classes, then at least one of them contains an arithmetic progression of length n. We prove in this paper that W(n) 2 2"/n " for all sufficiently large n. ..."
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The van der Waerden number W(n) is the smallest integer so that if we divide the integers {1,2,..., W(n)} into two classes, then at least one of them contains an arithmetic progression of length n. We prove in this paper that W(n) 2 2"/n " for all sufficiently large n.
A new method to construct lower bounds for van der Waerden numbers
 THE ELECTRONIC JOURNAL OF COMBINATORICS
, 2007
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A lower bound for offdiagonal van der Waerden numbers
 ADVANCES IN APPLIED MATHEMATICS
, 2010
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