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Van der Waerden's Theorem and Avoidability in Words [article]

Yu-Hin Au, Aaron Robertson, Jeffrey Shallit
2009 arXiv   pre-print
We consider some variations on this problem in the light of van der Waerden's theorem on arithmetic progressions.  ...  Theorem 8 was inspired by some related empirical calculations by Richard (Yang) Peng.  ...  Since this is a finite coloring of the positive integers, we know by van der Waerden's theorem that there exist n, d ≥ 1 such that Γ(n) = Γ(n + d) = . . . = Γ(n + αd).  ... 
arXiv:0812.2466v5 fatcat:7toejdgijjcflfie4wq5zhkm7e

On Applications of Van Der Waerden's Theorem

John R. Rabung, Van Der Waerden
1975 Mathematics Magazine  
An application of Witt's generalization of van der Waerden's theorem. Very shortly after van der Waerden published his proof in [2], A.  ...  ON APPLICATIONS OF VAN DER WAERDEN'S THEOREM JOHN R. RABUNG, Randolph-Macon College 1. Equivalent versions of the theorem. In [1] B.  ... 
doi:10.2307/2689695 fatcat:aw46att6svhydej2phhwko25nu

A note on van der Waerden's theorem

Alan D Taylor
1982 Journal of combinatorial theory. Series A  
In this note, we give a combinatorial version of the Furstenberg-Weiss topological proof of van der Waerden's theorem.  ...  From this, they easily derive the following celebrated result of van der Waerden: THEOREM (B. L. van der Waerden [3] ).  ... 
doi:10.1016/0097-3165(82)90011-5 fatcat:3h2p7rhpg5cmzb7shfvqcsmoru

On Applications of Van Der Waerden's Theorem

John R. Rabung
1975 Mathematics Magazine  
An application of Witt's generalization of van der Waerden's theorem. Very shortly after van der Waerden published his proof in [2], A.  ...  ON APPLICATIONS OF VAN DER WAERDEN'S THEOREM JOHN R. RABUNG, Randolph-Macon College 1. Equivalent versions of the theorem. In [1] B.  ... 
doi:10.1080/0025570x.1975.11976466 fatcat:kqm2f76tybf2tg5vbjfg45nsaq

Translation invariant filters and van der Waerden's Theorem [article]

Mauro Di Nasso
2020 arXiv   pre-print
We present a self-contained proof of a strong version of van der Waerden's Theorem.  ...  Notice that, as a straight consequence, one obtains the following strong version of van der Waerden's Theorem. Theorem 3.2.  ...  The following objects are the main ingredient in our proof of van der Waerden's Theorem. Definition 2.3.  ... 
arXiv:1808.01500v2 fatcat:yi2my2lfdfdh3mrupnlq55z644

Grünwald version of van der Waerden's theorem for semi-modules [article]

Xiongping Dai
2018 arXiv   pre-print
By establishing a Regional Multiple Recurrence Theorem for semimodules, we prove that one of the colors $j$ has the property that if $F\subseteq G$ is any finite set, then one can find some "syndetic"  ...  We can then obtain the following "Regional Multiple Recurrence Theorem" which generalizes and strengthens the classical theorem of Furstenberg and Weiss 1978 [18, Theorem 1.5] that is for the invertible  ...  version of van der Waerden's theorem as follows: Bergelson-Hindman Theorem ([7, Corollary 3.2]).  ... 
arXiv:1512.08695v4 fatcat:ctwcrdmznvbgdbfe7bqh5oan3a

A Canonical Polynomial Van der Waerden's Theorem [article]

António Girão
2020 arXiv   pre-print
We prove a canonical polynomial Van der Waerden's Theorem. More precisely, we show the following.  ...  Proof of the canonical Van Der Waerden's Theorem In this section, we give a sketch of a short proof of the Canonical Van der Waerden's Theorem.  ...  School A more recent and remarkable theorem in arithmetic Ramsey theory which once again extends Van der Waerden's Theorem is the polynomial Van der Waerden's Theorem, originally proved by Bergelson  ... 
arXiv:2004.07766v1 fatcat:rxrlnn4sy5g2no4flv3jmgyur4

Van der Waerden's Theorem and Avoidability in Words

Yu-Hin Au, Aaron Robertson, Jeffrey Shallit
2011 Integers  
Their tool was a famous one from combinatorics: namely, van der Waerden's theorem on arithmetic progressions [16] .  ...  Since this is a finite coloring of the positive integers, we know by van der Waerden's theorem that there exist n, d ≥ 1 such that Γ(n) = Γ(n + d) = . . . = Γ(n + αd).  ... 
doi:10.1515/integ.2011.007 fatcat:xvk5v3hkzzhjfhcuixmoipovtm

On a Variant of Van Der Waerden's Theorem

Sujith Vijay
2010 Integers  
Introduction A cornerstone of Ramsey theory is the theorem of van der Waerden [5] which states that for every positive integer k, there exists an integer W (k) such that any 2-coloring of {1, 2, . .  ...  Analogous to the van der Waerden number W (k), we can define Q n (k) as the least integer for which any 2-coloring of {1, 2, . . . , Q n (k)} yields a monochromatic k-term quasi-progression of diameter  ... 
doi:10.1515/integ.2010.017 fatcat:yufmrycqrfgujlrkeakln3zluu

On van der Waerden's theorem on arithmetic progressions

Walter Deuber
1982 Journal of combinatorial theory. Series A  
Proof of Theorem 1. Proceed by induction on 1. For I= 0 the theorem is obvious. Assume that the theorem holds for some 12 0 and arbitrary m. By induction on m prove it for 1 + 1.  ...  Theorem 1 may be viewed as the Hales-Jewett theorem rephrased in the language of arithmetic progressions. Thus in fact we have shown that this theorem may be proved using 2-colorings only.  ... 
doi:10.1016/0097-3165(82)90071-1 fatcat:mfhlyt7fnbdh5dcuv54xjbscmq

Superfilters, Ramsey theory, and van der Waerden's Theorem

Nadav Samet, Boaz Tsaban
2009 Topology and its Applications  
Superfilters are generalized ultrafilters, which capture the underlying concept in Ramsey theoretic theorems such as van der Waerden's Theorem.  ...  Following Bergelson and Hindman's 1989 Theorem, we present a new simultaneous generalization of the theorems of Ramsey, van der Waerden, Schur, Folkman-Rado-Sanders, Rado, and others, where the colored  ...  Back to van der Waerden Furstenberg-Weiss: AP (AP) 2 2 . AP is not even weakly Ramsey!  ... 
doi:10.1016/j.topol.2009.04.014 fatcat:mdik7jc2wnekpdev7za2uwb6sa

Variations on Van Der Waerden's and Ramsey's Theorems

T. C. Brown
1975 The American mathematical monthly  
Rabung, On applications of van der Waerden's theorem, Math. Magazine, 48 (1975) 142-148.  ...  Ramsey's Theorem is the following. Theorem R.  ...  Van der Waerden's Theorem The particular variation of van der Waerden's Theorem to be presented here has been discovered independently by any number of people, but a proof of its equivalence to van der  ... 
doi:10.1080/00029890.1975.11994010 fatcat:wbtvlvb5xna7vpbjrbqiftf7um

Van Der Waerden's Coloring Theorem and Classical Spin Systems

Ranjan Chaudhury, Debashis Gangopadhyay, Samir K. Paul
1997 Modern physics letters B  
We find a non-invertible matrix representation for Van der Waerden's colouring theorem for two distinct colours in a one dimensional periodic lattice.  ...  The authors would like to thank Sukumar Das Adhikary for introducing them to VDW Theorem.  ...  One of the many equivalent versions of Van der Waerden's Colouring Theorem in combinatoric number theory is 1) : Let X be a finite subset of N d .  ... 
doi:10.1142/s0217984997001134 fatcat:bgzspey7eja3fn6hsrv2lpxyhy

A sharp threshold for van der Waerden's theorem in random subsets

Mathias Schacht, Yury Person, Hiêp Hàn, Ehud Friedgut
2016 Discrete Analysis  
We establish sharpness for the threshold of van der Waerden's theorem in random subsets of $\mathbb{Z}/n\mathbb{Z}$.  ...  More precisely, for $k\geq 3$ and $Z\subseteq \mathbb{Z}/n\mathbb{Z}$ we say $Z$ has the van der Waerden property if any two-colouring of $Z$ yields a monochromatic arithmetic progression of length $k$  ...  In other words, we establish a sharp threshold for van der Waerden's theorem for two colours in Z{nZ.  ... 
doi:10.19086/da.615 fatcat:tuuql42aqrhgtl2buwqfrlptmi

A constructive topological proof of van der Waerden's theorem

Thierry Coquand
1995 Journal of Pure and Applied Algebra  
The theorem of van der Waerden on arithmetical progression is a classic result in combinatorics.  ...  While its original proof was of combinatorial nature, it was shown by Furstenberg and Weiss that this theorem can be derived from topological dynamics.  ...  Derivation of van der Waerden's theorem Before presenting the full derivation of van der Waerden's theorem, we find it illuminating to show the derivation that W'(3,l) is a covering for any fixed 1, which  ... 
doi:10.1016/0022-4049(94)00148-0 fatcat:m25fregb6bav7bk222auxqiqaq
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