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1. Remembrances

Author

Alon, Noga, primary, Brown, Tom C., additional, Butler, Steve, additional, Griggs, Jerrold R., additional, Hindman, Neil, additional, Jungic, Veselin, additional, Landman, Bruce M., additional, and Nešetřil, Jaroslav, additional

Published
2022
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2. New Bounds on van der Waerden-type Numbers for Generalized 3-term Arithmetic Progressions

Author

Allen, Patrick, Landman, Bruce M., and Meeks, Holly

Subjects
Mathematics - Combinatorics, 05D10
Abstract

Let a and b be positive integers with a \leq b. An (a,b)-triple is a set {x,ax+d,bx+ 2d}, where x,d \geq 1. Define T(a,b;r) to be the least positive integer n such that any r-coloring of {1,2...,n} contains a monochromatic (a,b)-triple. Earlier results gave an upper bound on T(a,b;2) that is a fourth degree polynomial in b and a, and a quadratic lower bound. A new upper bound for T(a,b;2) is given that is a quadratic. Additionally, lower bounds are given for the case in which a = b, updated tables are provided, and open questions are presented., Comment: The newer version reflects an updated table of values of T(a,b)

Published
2012

3. Bounds on Van der Waerden Numbers and Some Related Functions

Author

Brown, Tom, Landman, Bruce M., and Robertson, Aaron

Subjects
Mathematics - Combinatorics, 05D10
Abstract

For positive integers $s$ and $k_1, k_2, ..., k_s$, let $w(k_1,k_2,...,k_s)$ be the minimum integer $n$ such that any $s$-coloring $\{1,2,...,n\} \to \{1,2,...,s\}$ admits a $k_i$-term arithmetic progression of color $i$ for some $i$, $1 \leq i \leq s$. In the case when $k_1=k_2=...=k_s=k$ we simply write $w(k;s)$. That such a minimum integer exists follows from van der Waerden's theorem on arithmetic progressions. In the present paper we give a lower bound for $w(k,m)$ for each fixed $m$. We include a table with values of $w(k,3)$ which match this lower bound closely for $5 \leq k \leq 16$. We also give an upper bound for $w(k,4)$, an upper bound for $w(4;s)$, and a lower bound for $w(k;s)$ for an arbitrary fixed $k$. We discuss a number of other functions that are closely related to the van der Waerden function., Comment: 13 pages

Published
2007

4. Avoiding Monochromatic Sequences With Special Gaps

Author

Landman, Bruce M. and Robertson, Aaron

Subjects
Mathematics - Combinatorics, 05D10, 11B25, 11N13
Abstract

For $S$ a set of positive integers, and $k$ and $r$ fixed positive integers, denote by $f(S,k;r)$ the least positive integer $n$ (if it exists) such that within every $r$-coloring of $\{1,2,...,n\}$ there must be a monochromatic sequence $\{x_{1},x_{2},...,x_{k}\}$ with $x_{i}-x_{i-1} \in S$ for $2 \leq i \leq k$. We consider the existence of $f(S,k;r)$ for various choices of $S$, as well as upper and lower bounds on this function. In particular, we show that this function exists for all $k$ if $S$ is an odd translate of the set of primes and $r=2$., Comment: 16 pages

Published
2003

11. Ramsey Theory on the Integers

Author

Landman, Bruce M., Robertson, Aaron, Landman, Bruce M., and Robertson, Aaron

Subjects
Ramsey theory, Combinatorial analysis
Abstract

Ramsey theory is the study of the structure of mathematical objects that is preserved under partitions. In its full generality, Ramsey theory is quite powerful, but can quickly become complicated. By limiting the focus of this book to Ramsey theory applied to the set of integers, the authors have produced a gentle, but meaningful, introduction to an important and enticing branch of modern mathematics. Ramsey Theory on the Integers offers students a glimpse into the world of mathematical research and the opportunity for them to begin pondering unsolved problems. For this new edition, several sections have been added and others have been significantly updated. Among the newly introduced topics are: rainbow Ramsey theory, an “inequality” version of Schur's theorem, monochromatic solutions of recurrence relations, Ramsey results involving both sums and products, monochromatic sets avoiding certain differences, Ramsey properties for polynomial progressions, generalizations of the Erdős-Ginzberg-Ziv theorem, and the number of arithmetic progressions under arbitrary colorings. Many new results and proofs have been added, most of which were not known when the first edition was published. Furthermore, the book's tables, exercises, lists of open research problems, and bibliography have all been significantly updated. This innovative book also provides the first cohesive study of Ramsey theory on the integers. It contains perhaps the most substantial account of solved and unsolved problems in this blossoming subject. This breakthrough book will engage students, teachers, and researchers alike. Reviews of the Previous Edition: Students will enjoy it due to the highly accessible exposition of the material provided by the authors. —MAA Horizons What a wonderful book! … contains a very “student friendly” approach to one of the richest areas of mathematical research … a very good way of introducing the students to mathematical research … an extensive bibliography … no other book on the subject … which is structured as a textbook for undergraduates … The book can be used in a variety of ways, either as a textbook for a course, or as a source of research problems … strongly recommend this book for all researchers in Ramsey theory … very good book: interesting, accessible and beautifully written. The authors really did a great job! —MAA Online

Published
2015

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