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Ramsey functions related to the van der waerden numbers

Bruce M. Landman
1992 Discrete Mathematics  
., Ramsey functions related to the van der Waerden numbers, Discrete Mathematics 102 (1992) 265-278.  ...  Introduction One of the more difficult problems in Ramsey theory has been the estimation of the van der Waerden numbers.  ...  If, in the definition of w(n), we substitute for the class 9 of arithmetic progressions, a class 9', containing 9, and define w'(n) to be the Ramsey number corresponding to 9' instead of 9, it is clear  ... 
doi:10.1016/0012-365x(92)90120-5 fatcat:splyrwhsbvdmtjy73tai7lb5jq

On Ramsey-type positional games

Jaroslav Nešetřil, Tomáš Valla
2009 Journal of Graph Theory  
Beck introduced the concept of Ramsey-type games by studying the game versions of Ramsey's and van der Waerden's theorems.  ...  We contribute to this topic by investigating games corresponding to structural extensions of Ramsey's and van der Waerden's theorems-the theorem of Brauer, structural and restricted Ramsey theorems.  ...  For Theorem 1, let us just recall that the Van der Waerden function is known to be primitive recursive (as shown first by Shelah [11] ) and the two tower function bound was obtained recently by Gowers  ... 
doi:10.1002/jgt.20463 fatcat:hahj7n36czfmnla36shz2sg2be

Satisfiability and computing van der Waerden numbers [article]

Michael R. Dransfield, Victor W. Marek, Miroslaw Truszczynski
2003 arXiv   pre-print
By following it, we were able to obtain several new results pertaining to the problem of computing van der Waerden numbers.  ...  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  ...  During the research reported in this paper the second and third authors have been partially supported by an NSF grant IIS-0097278.  ... 
arXiv:cs/0310064v1 fatcat:5qaphuqlzvhlzaklxxhhuwaunq

Forthcoming papers

1991 Discrete Mathematics  
Landman, Ramsey functions related to the van der Waerden numbers Ramsey functions similar to the van der Waerden numbers w(n) are studied.  ...  Henning, Bounds relating generalized domination parameters The domination number y(G) and the total domination number y,(G) of a graph G are generalized to the &-domination number y&G) and the total K,  ... 
doi:10.1016/0012-365x(91)90360-e fatcat:le2sw7ugjvg5nf6zbtmwcnyh6u

Ramsey Theory in the Work of Paul Erdős [chapter]

Ron L. Graham, Jaroslav Nešetřil
2013 The Mathematics of Paul Erdős II  
But perhaps one could say that Ramsey theory was created largely by him. This paper will attempt to demonstrate this claim.  ...  van der Waerden numbers.  ...  Choosing pix = x and i = i we get the van der Waerden theorem.  ... 
doi:10.1007/978-1-4614-7254-4_13 fatcat:vqvyhfhj3ngr7c3vokrsmrt2ai

Ramsey Theory in the Work of Paul Erdős [chapter]

R. L. Graham, J. Nešetřil
1997 Algorithms and Combinatorics  
But perhaps one could say that Ramsey theory was created largely by him. This paper will attempt to demonstrate this claim.  ...  van der Waerden numbers.  ...  Choosing pix = x and i = i we get the van der Waerden theorem.  ... 
doi:10.1007/978-3-642-60406-5_16 fatcat:cqkc3inzjfe4lbu2jj4oem2fv4

On efficient constructions of short lists containing mostly Ramsey graphs [article]

Marius Zimand
2012 arXiv   pre-print
One of the earliest and best-known application of the probabilistic method is the proof of existence of a 2 log n-Ramsey graph, i.e., a graph with n nodes that contains no clique or independent set of  ...  The explicit construction of such a graph is a major open problem.  ...  Constructive lower bounds for the van der Waerden theorem Van der Waerden Theorem is another classical result in Ramsey theory.  ... 
arXiv:1210.4408v1 fatcat:l2prdbgqxbfw5ottty55sg6bl4

On Efficient Constructions of Short Lists Containing Mostly Ramsey Graphs [chapter]

Marius Zimand
2013 Lecture Notes in Computer Science  
One of the earliest and best-known application of the probabilistic method is the proof of existence of a 2 log n-Ramsey graph, i.e., a graph with n nodes that contains no clique or independent set of  ...  The explicit construction of such a graph is a major open problem.  ...  Constructive Lower Bounds for the van der Waerden Theorem Van der Waerden Theorem is another classical result in Ramsey theory.  ... 
doi:10.1007/978-3-642-38236-9_19 fatcat:edfzvksbqnbkngsdamanhn5ifa

On the van der Waerden numbers w(2;3,t)

Tanbir Ahmed, Oliver Kullmann, Hunter Snevily
2014 Discrete Applied Mathematics  
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).  ...  Different from the situation for ordinary van der Waerden numbers, these "numbers" need actually to be pairs of numbers.  ...  Acknowledgements The authors would like to thank Donald Knuth, the Editor and the anonymous referees for their valuable suggestions and helpful comments.  ... 
doi:10.1016/j.dam.2014.05.007 fatcat:g76nnnpspfd3rn64nvydmx6cuy

A new lower bound for van der Waerden numbers

Thomas Blankenship, Jay Cummings, Vladislav Taranchuk
2018 European journal of combinatorics (Print)  
The recurrence can also be used to construct explicit valid colorings, and it improves known lower bounds on small van der Waerden numbers.  ...  In this paper we prove a new recurrence relation on the van der Waerden numbers, w(r,k). In particular, if p is a prime and p≤ k then w(r, k) > p ·(w(r - r/p, k) -1).  ...  Acknowledgments The authors would like to thank Craig Timmons for his helpful comments which improved this paper. Keywords: Van der Waerden numbers.  ... 
doi:10.1016/j.ejc.2017.10.007 fatcat:evegowekaraapnebvys3rsmfq4

The van der Waerden complex

Richard Ehrenborg, Likith Govindaiah, Peter S. Park, Margaret Readdy
2017 Journal of Number Theory  
We introduce the van der Waerden complex vdW(n,k) defined as the simplicial complex whose facets correspond to arithmetic progressions of length k in the vertex set {1, 2, ..., n}.  ...  We show the van der Waerden complex vdW(n,k) is homotopy equivalent to a CW-complex whose cells asymptotically have dimension at most k / k.  ...  Acknowledgments The authors thank Nigel Pitt for discussions related to asymptotics in Section 3. The first author was partially supported by National Security Agency grant H98230-13-1-0280.  ... 
doi:10.1016/j.jnt.2016.08.012 fatcat:vm6zuc3bobhk5frlik56q37si4

The Deluge of Spurious Correlations in Big Data

Cristian S. Calude, Giuseppe Longo
2016 Foundations of Science  
Consequently, there will be no need to give scientific meaning to phenomena, by proposing, say, causal relations, since regularities in very large databases are enough: "with enough data, the numbers speak  ...  The "end of science" is proclaimed. Using classical results from ergodic theory, Ramsey theory and algorithmic information theory, we show that this "philosophy" is wrong.  ...  Vulpiani and the anonymous referees for useful comments and suggestions.  ... 
doi:10.1007/s10699-016-9489-4 fatcat:jky3r56oqzg6didppz7uz7mf6i

Multi-recurrence and van der Waerden systems

Dominik Kwietniak, Jian Li, Piotr Oprocha, XiangDong Ye
2016 Science China Mathematics  
We explore recurrence properties arising from dynamical approach to combinatorial problems like the van der Waerden Theorem.  ...  We describe relations between these properties, study their consequences for dynamics, and demonstrate connections to combinatorial problems.  ...  In this section, we study this property and its relation to the van der Waerden systems. Definition 5.1.  ... 
doi:10.1007/s11425-015-0860-8 fatcat:bc5qoeboxrfjhdp37ia2t7iedu

Some New Exact van der Waerden Numbers [article]

Bruce Landman, Aaron Robertson, Clay Culver
2005 arXiv   pre-print
,k_r-1, the van der Waerden number w(k_0,k_1,...,k_r-1) is the least positive integer n such that whenever {1,2,...,n} is partitioned into r sets S_0,S_1,...  ...  In addition, for the situation in which only one value of k_i differs from 2, we give a precise formula for the van der Waerden function (provided this one value of k_i is not too small)  ...  Acknowledgement We thank Carl Pomerance for pointing us to some relevant literature.  ... 
arXiv:math/0507019v1 fatcat:lb52wutsbfcmxpxh5m2esapm7u

Shelah's partition functions and the Hales-Jewett numbers [article]

Mohammad Golshani, Mostafa Mirabi
2021 arXiv   pre-print
In this paper we study several partition relations, defined by Saharon Shelah, and relate them to the Hales-Jewett numbers.  ...  In particular we give an upper bound for the Hales-Jewett numbers using the primitive recursive function 𝚏^8,* which belongs to the class ℰ^5 of the Grzegorczyk hierarchy and grows slower than the function  ...  The authors also would like to thank Alex Kruckman for his very careful reading of the paper and for the helpful comments and corrections.  ... 
arXiv:2104.05962v2 fatcat:fjsixpbo55dsjeottsc3ayoq5e
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