Bounds on some van der Waerden numbers

Brown, Tom C.; Landman, Bruce; Robertson, Aaron

doi:10.1016/j.jcta.2008.01.005

Cited by 14 publications

(26 citation statements)

References 11 publications

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“…The numbers vdw 2 (3, k) are known for 1 ≤ k ≤ 18 (see above). Additionally, our experiments yield the following conjectured values (where using "≥ x" means that we believe that actually equality holds, while the lower bound "> x − 1" has been shown), further supporting the conjectured upper bound: 5) (4,9) > 254 is in [1]; we can improve this to vdw 2 (4, 9) ≥ 309, and furthermore vdw 2 (4, 10) > 328. So going from k = 8 to k = 9 we see a rather big jump, however possibly from k = 9 to k = 10 only a small change might take place.…”

Section: Definitionsupporting

confidence: 74%

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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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Section: Definitionsupporting

confidence: 74%

“…3. For general t = (k 0 ) with k 0 ≥ 3, in [5] the lower bound vdw 2 (k 0 , k) ≥ k k0−1−log(log(k)) for sufficiently large k has been shown. It seems consistent with current knowledge that we could have vdw 2 (k 0 , k) ≤ k k0−1 for all k, k 0 ≥ 1.…”

Section: Definitionmentioning

confidence: 99%

Green-Tao Numbers and SAT

Kullmann

2010

Theory and Applications of Satisfiability Testing – SAT 2010

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“…The known values [4] are w 1 .2I 3; 2/ D 9, w 1 .2I 3; 3/ D 23, w 1 .2I 3; 4/ D 34, w 1 .2I 3; 5/ D 73, w 1 .2I 3; 6/ D 113 and w 1 .2I 3; 7/ D 193. Lower bounds of w 1 .2I 3; k/ can be found in [9].…”

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confidence: 99%

Some new van der Waerden numbers and some van der Waerden-type numbers

Ahmed¹

2009

Integers

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The van der Waerden number w.rI k 1 ; k 2 ; : : : ; k r / is the least m such that given any partition ¹1; 2; : : : ; mº D P 1 [ P 2 [ [ P r , there is an index j 2 ¹1; 2; : : : ; rº such that P j contains an arithmetic progression of length k j . We have computed exact values of some (30) previously unknown van der Waerden numbers and also computed lower bounds of others. Let w d .rI k 1 ; k 2 ; : : : ; k r / be the least m such that given any partition ¹1; 2; : : : ; mº D P 1 [ P 2 [ [ P r , there is an index j 2 ¹1; 2; : : : ; r 1º such that P j contains an arithmetic progression of length k j , or P r contains an arithmetic progression of length k r with common difference at most d . A table of observed values of w d .rI k 1 ; k 2 ; : : : ; k r / for d D 1; 2; : : : ; b m 1 kr 1 c is given.

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“…where: (3) holds since there is exactly one such k-ap that intersects b in k − 1 − i places and Lemma 6(i) gives i ≥ 2; (4) follows from Lemma 5(ii) and Lemma 6(iii); (5) holds from the lower bound in Lemma 5(ii) and since 3sk or fewer k-aps intersect a given k-ap (this is a standard bound typically used with the Lovasz Local Lemma; see, e.g., [2]); and (6) holds by using the upper bound in (2) along with independence since the two k-aps do not share any element.…”

Section: Resultsmentioning

confidence: 99%