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Van der Waerden's Theorem on Homothetic Copies of {1,1+s,1+s+t}
-
Byeong Moon Kim
and
Yoomi Rho
Published/Copyright:
October 2, 2012
Abstract.
For all positive
integers s and t, Brown et al. defined 
to be
the smallest positive integer N such that every 2-coloring of
has a monochromatic homothetic copy of {1,1+s,1+s+t}. They proved that 
for all s, t and
that equality holds in the case where both
and 
with
and in many other cases. They also proved that for all positive integers m, 
or 
. In this paper, we show that and that for
all the other (s,t), 
.
Keywords: Van der Waerden's Theorem; Arithmetic Progressions; Homothetic Copies; 2-Colorings; Monochromatic Triples
Received: 2011-08-17
Revised: 2012-02-02
Accepted: 2012-03-12
Published Online: 2012-10-02
Published in Print: 2012-10-01
© 2012 by Walter de Gruyter Berlin Boston
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Articles in the same Issue
- Masthead
- Two-Color Babylon
- On Abelian and Additive Complexity in Infinite Words
- Approximations of Additive Squares in Infinite Words
- The Googol-th Bit of the Erdős–Borwein Constant
- Odd Repdigits to Small Bases Are Not Perfect
- Enumeration of the Degree Sequences of Line-Hamiltonian Multigraphs
- The Characteristic Sequence and p-Orderings of the Set of d-th Powers of Integers
- Van der Waerden's Theorem on Homothetic Copies of {1,1+s,1+s+t}
- Sum-Products Estimates with Several Sets and Applications
- The 2-Adic, Binary and Decimal Periods of 1/3k Approach Full Complexity for Increasing k
- A Proof of Catalan's Convolution Formula
- Explicit Constructions of Large Families of Generalized More Sums Than Differences Sets
- The Impossibility of Certain Types of Carmichael Numbers
- Integer Subsets with High Volume and Low Perimeter
- Mean-Value Theorems for Multiplicative Arithmetic Functions of Several Variables
- A Generalized Diagonal Wythoff Nim
- A Recursive Process Related to a Partizan Variation of Wythoff
- The (r1,...,rp)-Stirling Numbers of the Second Kind
- Factor Frequencies in Languages Invariant Under Symmetries Preserving Factor Frequencies
- On k-Lehmer Numbers
Keywords for this article
Van der Waerden's Theorem;
Arithmetic Progressions;
Homothetic Copies;
2-Colorings;
Monochromatic Triples
Articles in the same Issue
- Masthead
- Two-Color Babylon
- On Abelian and Additive Complexity in Infinite Words
- Approximations of Additive Squares in Infinite Words
- The Googol-th Bit of the Erdős–Borwein Constant
- Odd Repdigits to Small Bases Are Not Perfect
- Enumeration of the Degree Sequences of Line-Hamiltonian Multigraphs
- The Characteristic Sequence and p-Orderings of the Set of d-th Powers of Integers
- Van der Waerden's Theorem on Homothetic Copies of {1,1+s,1+s+t}
- Sum-Products Estimates with Several Sets and Applications
- The 2-Adic, Binary and Decimal Periods of 1/3k Approach Full Complexity for Increasing k
- A Proof of Catalan's Convolution Formula
- Explicit Constructions of Large Families of Generalized More Sums Than Differences Sets
- The Impossibility of Certain Types of Carmichael Numbers
- Integer Subsets with High Volume and Low Perimeter
- Mean-Value Theorems for Multiplicative Arithmetic Functions of Several Variables
- A Generalized Diagonal Wythoff Nim
- A Recursive Process Related to a Partizan Variation of Wythoff
- The (r1,...,rp)-Stirling Numbers of the Second Kind
- Factor Frequencies in Languages Invariant Under Symmetries Preserving Factor Frequencies
- On k-Lehmer Numbers
