Two New Van Der Waerden Numbers: w(2; 3, 17) and w(2; 3, 18)

Ahmed, Tanbir

doi:10.1515/integ.2010.032

Cited by 11 publications

(27 citation statements)

References 6 publications

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“…By a good partition, we mean a partition of the form (1) such that no P j contains an arithmetic progression of t j terms. We have recently published some previously unknown van der Waerden numbers in [1,2]. 418 T.…”

Section: Introductionmentioning

confidence: 99%

“…418 T. Ahmed To make this article self-contained, we repeat some definitions that we provided in [2]. We construct an instance F of the satisfiability problem (described in the following paragraph) with n variables for w.kI t 0 ; t 1 ; : : : ; t k 1 / such that F is satisfiable if and only if n < w.kI t 0 ; t 1 ; : : : ; t k 1 /.…”

Section: Introductionmentioning

confidence: 99%

“…Our implementation is engineered to run faster on unsatisfiable instances corresponding to van der Waerden numbers. For a brief description on this implementation, see Section 3 of [2].…”

Section: Introductionmentioning

confidence: 99%

“…For a brief description, see Section 4 of [2]. In [2], we have reported the exact values of w.2I 3; 17/ and w.2I 3; 18/ to be 279 and 312, respectively.…”

Section: Introductionmentioning

confidence: 99%

“…For a brief description, see Section 4 of [2]. In [2], we have reported the exact values of w.2I 3; 17/ and w.2I 3; 18/ to be 279 and 312, respectively. In Section 2 of this article, we report the exact value of w.2I 4; 9/ and lower bounds of w.2I 5; t 1 / with 7 6 t 1 6 11.…”

Section: Introductionmentioning

confidence: 99%

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On Computation of Exact van der Waerden Numbers

Ahmed

2012

Integers

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The van der Waerden number w.kI t 0 ; t 2 ; : : : ; t k 1 / is the smallest integer m such that in every partition P 0 [ P 1 [ [ P k 1 of the set ¹1; 2; : : : ; mº there is always a block P j that contains an arithmetic progression of length t j . In this paper, we report the exact value of the previously unknown van der Waerden number w.2I 4; 9/, some lower bounds of w.2I 5; t/ and polynomial upper-bound conjectures for w.2I 4; t / and w.2I 5; t /. We also present an efficient SAT-encoding of the number w.kI t 0 ; t 1 ; : : : ; t k 1 / for k > 3 using which we have computed the exact value of w.3I 3; 3; 6/ and some lower bounds of w.3I t 0 ; t 1 ; t 2 /.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Our implementation is engineered to run faster on unsatisfiable instances corresponding to van der Waerden numbers. For a brief description on this implementation, see Section 3 of [2].…”

Section: Introductionmentioning

confidence: 99%

“…For a brief description, see Section 4 of [2]. In [2], we have reported the exact values of w.2I 3; 17/ and w.2I 3; 18/ to be 279 and 312, respectively.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On Computation of Exact van der Waerden Numbers

Ahmed

2012

Integers

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Solving and Verifying the Boolean Pythagorean Triples Problem via Cube-and-Conquer

Heule

Kullmann

Marek

2016

Lecture Notes in Computer Science

144

110

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Abstract. The boolean Pythagorean Triples problem has been a longstanding open problem in Ramsey Theory: Can the set N = {1, 2, . . . } of natural numbers be divided into two parts, such that no part contains a triple (a, b, c) with a 2 + b 2 = c 2 ? A prize for the solution was offered by Ronald Graham over two decades ago. We solve this problem, proving in fact the impossibility, by using the Cube-and-Conquer paradigm, a hybrid SAT method for hard problems, employing both look-ahead and CDCL solvers. An important role is played by dedicated look-ahead heuristics, which indeed allowed to solve the problem on a cluster with 800 cores in about 2 days. Due to the general interest in this mathematical problem, our result requires a formal proof. Exploiting recent progress in unsatisfiability proofs of SAT solvers, we produced and verified a proof in the DRAT format, which is almost 200 terabytes in size. From this we extracted and made available a compressed certificate of 68 gigabytes, that allows anyone to reconstruct the DRAT proof for checking.

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Green-Tao Numbers and SAT

Kullmann

2010

Theory and Applications of Satisfiability Testing – SAT 2010

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Abstract. We consider the links between Ramsey theory in the integers, based on van der Waerden's theorem, and (boolean, CNF) SAT solving. We aim at using the problems from exact Ramsey theory, concerned with computing Ramsey-type numbers, as a rich source of test problems, where especially methods for solving hard problems can be developed. We start our investigations here by reviewing the known van der Waerden numbers, and we discuss directions in the parameter space where possibly the growth of van der Waerden numbers vdwm(k1, . . . , km) is only polynomial (this is important for obtaining feasible problem instances). We introduce transversal extensions as a natural way of constructing mixed parameter tuples (k1, . . . , km) for van-der-Waerden-like numbers N(k1, . . . , km), and we show that the growth of the associated numbers is guaranteed to be linear. Based on Green-Tao's theorem ("the primes contain arbitrarily long arithmetic progressions") we introduce the GreenTao numbers grt m (k1, . . . , km), which in a sense combine the strict structure of van der Waerden problems with the (pseudo-)randomness of the distribution of prime numbers. Using standard SAT solvers (look-ahead, conflict-driven, and local search) we determine the basic values. It turns out that already for this single form of Ramsey-type problems, when considering the best-performing solvers a wide variety of solver types is covered. For m > 2 the problems are non-boolean, and we introduce the generic translation scheme, which offers an infinite variety of translations ("encodings") and covers the known methods. In most cases the special instance called nested translation proved to be far superior over its competitors (including the direct translation). IntroductionThe applicability of SAT solvers has made tremendous progress over the last 15 years; see the recent handbook [3]. We are concerned here with solving (concrete) combinatorial problems (see [33] for an overview). Especially we are concerned 2 with the computation of van-der-Waerden-like numbers, which is about colouring hypergraphs of arithmetic progressions. 1)An arithmetic progression of size k ∈ N 0 in N is a set P ⊂ N of size k such that after ordering (in the natural order), two neighbours always have the same distance. So the arithmetic progressions of size k > 1 are the sets of the form P = {a + i · d : i ∈ {0, . . . , k − 1}} for a, d ∈ N. Van der Waerden's Theorem ([32]) shows that whenever the set N of natural numbers is partitioned into finitely many parts, some part must contain arithmetic progressions of arbitrary size. The finite version, which is equivalent to the above infinite version, says that for every progression size k ∈ N and every number m ∈ N of parts there exists some n 0 ∈ N such that for n ≥ n 0 every partitioning of {1, . . . , n} into m parts has some part which contains an arithmetic progression of size k. The smallest such n 0 is denoted by vdw m (k), and is called a vdW-number. The subfield of Ramsey theory concerned with van der Waerden's theorem...

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Two New Van Der Waerden Numbers: w(2; 3, 17) and w(2; 3, 18)

Cited by 11 publications

References 6 publications

On Computation of Exact van der Waerden Numbers

On Computation of Exact van der Waerden Numbers

Solving and Verifying the Boolean Pythagorean Triples Problem via Cube-and-Conquer

Green-Tao Numbers and SAT

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