1,223 research outputs found
Bounds on some van der Waerden numbers
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 s-coloring of set {1, 2, . . . , n}, with colors 1, 2, . . . , s, there is a k i -term arithmetic progression of color i for some i. We give an asymptotic lower bound for w(k, m; 2) for fixed m. We include a table of values of w(k, 3; 2) that are very close to this lower bound for m = 3. We also give a lower bound for w(k, k, . . . , k; s) that slightly improves previously-known bounds. Upper bounds for w(k, 4; 2) and w(4, 4, . . . , 4; s) are also provided
New Lower Bounds for van der Waerden Numbers Using Distributed Computing
This paper provides new lower bounds for van der Waerden numbers. The number
is defined to be the smallest integer for which any -coloring
of the integers admits monochromatic arithmetic progression of
length ; its existence is implied by van der Waerden's Theorem. We exhibit
-colorings of that do not contain monochromatic arithmetic
progressions of length to prove that . These colorings are
constructed using existing techniques. Rabung's method, given a prime and a
primitive root , applies a color given by the discrete logarithm base
mod and concatenates copies. We also used Herwig et al's
Cyclic Zipper Method, which doubles or quadruples the length of a coloring,
with the faster check of Rabung and Lotts. We were able to check larger primes
than previous results, employing around 2 teraflops of computing power for 12
months through distributed computing by over 500 volunteers. This allowed us to
check all primes through 950 million, compared to 10 million by Rabung and
Lotts. Our lower bounds appear to grow roughly exponentially in . Given that
these constructions produce tight lower bounds for known van der Waerden
numbers, this data suggests that exact van der Waerden Numbers grow
exponentially in with ratio asymptotically, which is a new conjecture,
according to Graham.Comment: 8 pages, 1 figure. This version reflects new results and reader
comment
On the van der Waerden numbers w(2;3,t)
We present results and conjectures on the van der Waerden numbers w(2;3,t)
and on the new palindromic van der Waerden numbers pdw(2;3,t). We have computed
the new number w(2;3,19) = 349, and we provide lower bounds for 20 <= t <= 39,
where for t <= 30 we conjecture these lower bounds to be exact. The lower
bounds for 24 <= t <= 30 refute the conjecture that w(2;3,t) <= t^2, and we
present an improved conjecture. We also investigate regularities in the good
partitions (certificates) to better understand the lower bounds.
Motivated by such reglarities, we introduce *palindromic van der Waerden
numbers* pdw(k; t_0,...,t_{k-1}), defined as ordinary van der Waerden numbers
w(k; t_0,...,t_{k-1}), however only allowing palindromic solutions (good
partitions), defined as reading the same from both ends. Different from the
situation for ordinary van der Waerden numbers, these "numbers" need actually
to be pairs of numbers. We compute pdw(2;3,t) for 3 <= t <= 27, and we provide
lower bounds, which we conjecture to be exact, for t <= 35.
All computations are based on SAT solving, and we discuss the various
relations between SAT solving and Ramsey theory. Especially we introduce a
novel (open-source) SAT solver, the tawSolver, which performs best on the SAT
instances studied here, and which is actually the original DLL-solver, but with
an efficient implementation and a modern heuristic typical for look-ahead
solvers (applying the theory developed in the SAT handbook article of the
second author).Comment: Second version 25 pages, updates of numerical data, improved
formulations, and extended discussions on SAT. Third version 42 pages, with
SAT solver data (especially for new SAT solver) and improved representation.
Fourth version 47 pages, with updates and added explanation
Using the Incompressibility Method to obtain Local Lemma results for Ramsey-type Problems
We reveal a connection between the incompressibility method and the Lovasz
local lemma in the context of Ramsey theory. We obtain bounds by repeatedly
encoding objects of interest and thereby compressing strings. The method is
demonstrated on the example of van der Waerden numbers. It applies to lower
bounds of Ramsey numbers, large transitive subtournaments and other Ramsey
phenomena as well.Comment: 8 pages, 1 figur
Lower Bounds on the van der Waerden Numbers: Randomized- and Deterministic-Constructive
The van der Waerden number W(k,2) is the smallest integer n such that every
2-coloring 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 developing lower bounds
on W(k,2). Most of these lower bound proofs employ the probabilistic method
often in combination with the Lov\'asz Local Lemma. While these proofs show the
existence of a 2-coloring that has no monochromatic arithmetic progression of
length k they provide no efficient algorithm to find such a coloring. These
kind of proofs are often informally called nonconstructive in contrast to
constructive proofs that provide an efficient algorithm.
This paper clarifies these notions and gives definitions for deterministic-
and randomized-constructive proofs as different types of constructive proofs.
We then survey the literature on lower bounds on W(k,2) in this light. We show
how known nonconstructive lower bound proofs based on the Lov\'asz Local Lemma
can be made randomized-constructive using the recent algorithms of Moser and
Tardos. We also use a derandomization of Chandrasekaran, Goyal and Haeupler to
transform these proofs into deterministic-constructive proofs. We provide
greatly simplified and fully self-contained proofs and descriptions for these
algorithms
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