Glossary

5.2 Van der Waerden numbers and their properties

2 min read•july 25, 2024

Van der Waerden numbers are crucial in Ramsey theory, guaranteeing monochromatic arithmetic progressions in colored sets. They're denoted as w(k,r), where k is the progression length and r is the number of colors used.

Calculating these numbers is tough, with exact values known only for small k and r. They grow rapidly, bounded by complex formulas. Properties like and help understand their behavior, but many mysteries remain unsolved.

Van der Waerden Numbers: Definition and Basic Properties

Definition of Van der Waerden numbers

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  • Van der Waerden numbers denoted as [w(k,r)](https://www.fiveableKeyTerm:w(k,r))[w(k, r)](https://www.fiveableKeyTerm:w(k,_r)) represent smallest positive integer nn guaranteeing of length kk for any rr- of {1,2,...,n}\{1, 2, ..., n\}

  • kk (length of arithmetic progression) and rr (number of colors) determine specific

  • Significance in Ramsey theory guarantees existence of certain structures in colored sets and relates to on arithmetic progressions

Calculation of Van der Waerden numbers

  • Small Van der Waerden numbers include [w(3,2)](https://www.fiveableKeyTerm:w(3,2))=9[w(3, 2)](https://www.fiveableKeyTerm:w(3,_2)) = 9, [w(4,2)](https://www.fiveableKeyTerm:w(4,2))=35[w(4, 2)](https://www.fiveableKeyTerm:w(4,_2)) = 35, and [w(5,2)](https://www.fiveableKeyTerm:w(5,2))=178[w(5, 2)](https://www.fiveableKeyTerm:w(5,_2)) = 178

  • Growth rate increases rapidly with kk and rr, bounded above by w(k,r)≤2222(k−1)⋅rw(k, r) \leq 2^{2^{2^{2^{(k-1)}}}}\cdot r and below by w(k,r)≥rk−1w(k, r) \geq r^{k-1}

  • Computational challenges arise as exact values known only for small kk and rr, becoming increasingly difficult to calculate for larger parameters ()

Properties of Van der Waerden numbers

  • Monotonicity states if k1≤k2k_1 \leq k_2 and r1≤r2r_1 \leq r_2, then w(k1,r1)≤w(k2,r2)w(k_1, r_1) \leq w(k_2, r_2), as larger parameters require more elements to guarantee desired structure

  • Subadditivity property w(k,r1+r2)≤w(k,r1)+w(k,r2)−1w(k, r_1 + r_2) \leq w(k, r_1) + w(k, r_2) - 1 proved by combining separate colorings and applying

  • Symmetry w(k,r)=w(r,k)w(k, r) = w(r, k) holds for k=2k = 2, relating to Ramsey numbers through inequality w(k,r)≤R(k,k,...,k)w(k, r) \leq R(k, k, ..., k) (rr times)

Bounds for Van der Waerden numbers

  • Upper bounds include w(k,r)≤2222(k−1)⋅rw(k, r) \leq 2^{2^{2^{2^{(k-1)}}}}\cdot r and w(k,2)≤2222k+9w(k, 2) \leq 2^{2^{2^{2^{k+9}}}}

  • Lower bounds derived from w(p+1,2)≥p⋅2pw(p + 1, 2) \geq p \cdot 2^p for prime pp and w(k,r)≥rk−1w(k, r) \geq r^{k-1}

  • Large discrepancy between upper and lower bounds highlights gaps in knowledge, with exact values unknown for most kk and rr

  • Recent developments improve bounds for specific cases and establish connections to other areas of mathematics (additive combinatorics) and computer science (complexity theory)

Key Terms to Review (21)

Berlekamp's Construction: Berlekamp's Construction is a method used in combinatorial mathematics, specifically in Ramsey Theory, to construct a specific type of coloring of the integers. This construction is particularly important when analyzing Van der Waerden numbers, as it demonstrates the existence of certain monochromatic arithmetic progressions within any coloring of the integers. By systematically analyzing these colorings, Berlekamp's Construction reveals properties related to the stability and thresholds of Van der Waerden numbers.
Bertand van der Waerden: Bertand van der Waerden was a prominent Dutch mathematician known for his significant contributions to combinatorics and number theory, particularly through his formulation of Van der Waerden's Theorem. This theorem establishes that for any given integers, there exists a certain minimum size of a set of natural numbers such that if the set is colored with a finite number of colors, it will contain monochromatic arithmetic progressions. This result connects deeply with Van der Waerden numbers and their properties, which quantify the minimum set size needed to guarantee such progressions in specific coloring scenarios.
Coloring: In combinatorial mathematics, coloring refers to the assignment of labels or colors to elements of a set in such a way that certain conditions are met. This concept plays a crucial role in Ramsey Theory, particularly when analyzing relationships within graphs or sets, helping to uncover patterns and structure related to combinatorial problems.
Combinatorial Optimization: Combinatorial optimization is a field of mathematical optimization that focuses on finding the best solution from a finite set of possible solutions. It often deals with problems involving discrete structures, where the goal is to maximize or minimize a particular objective function under given constraints. This concept connects to various applications, including graph theory, resource allocation, and scheduling problems, linking it to broader themes in mathematics and computer science.
Exponential Lower Bound: An exponential lower bound is a type of mathematical estimate that indicates the minimum possible growth rate or size of a mathematical object, often expressed as a function that grows exponentially with respect to some variable. In the context of Van der Waerden numbers, it shows that these numbers grow very rapidly and provides insight into the inherent complexity and structure of combinatorial problems. Understanding exponential lower bounds helps in analyzing how certain configurations are unavoidable in large enough systems.
Gowers' bound: Gowers' bound refers to a mathematical result concerning the growth of the van der Waerden numbers, which are crucial in Ramsey Theory. It provides an upper bound for these numbers, indicating how large they can become based on the number of colors and the length of the arithmetic progression sought. This result helps in understanding the complexity and structure of combinatorial problems related to coloring and sequences.
Graham-Rothschild Theorem: The Graham-Rothschild Theorem is a result in Ramsey Theory that generalizes the classic Ramsey's Theorem. It states that for any partition of the n-dimensional hypercube into finitely many classes, there exists a large enough dimension such that one of the classes contains a large structured subset. This theorem connects deeply with concepts such as Van der Waerden numbers, the Hales-Jewett Theorem, and various properties of parameter sets.
Graph Theory: Graph theory is a branch of mathematics that studies the properties and relationships of graphs, which are structures made up of vertices (or nodes) connected by edges. This mathematical framework allows for the exploration of various problems across numerous fields, including combinatorics, computer science, and network theory, providing tools to analyze and understand complex structures and their interactions.
Monochromatic arithmetic progression: A monochromatic arithmetic progression is a sequence of numbers in which the terms are evenly spaced and all belong to the same color or category, typically in a coloring of integers. This concept is central to understanding how numbers can be arranged under certain constraints and is fundamental in exploring combinatorial structures, particularly regarding the existence and properties of these progressions within various colorings.
Monotonicity: Monotonicity refers to the property of a function or sequence that consistently increases or decreases, without any fluctuations. This concept is vital in understanding the behavior of functions within combinatorial contexts, particularly in determining patterns and relationships between elements, as seen in the study of specific numbers and theorems in Ramsey Theory.
Np-hard problem: An np-hard problem is a class of problems that are at least as hard as the hardest problems in NP (nondeterministic polynomial time). This means that if any np-hard problem can be solved in polynomial time, then every problem in NP can also be solved in polynomial time. These problems often relate to combinatorial optimization and, when connected to Ramsey Theory, illustrate how complex it can be to find specific configurations or solutions within large sets.
Parameters: Parameters are specific numerical values or variables that define the conditions of a mathematical problem or framework. In the context of Van der Waerden numbers, parameters are crucial for determining the configurations of colors and the lengths of sequences that lead to guaranteed monochromatic subsequences. These parameters help in exploring the relationships between different aspects of combinatorial structures and coloring problems.
Paul Erdős: Paul Erdős was a prolific Hungarian mathematician known for his extensive contributions to number theory, combinatorics, and graph theory, particularly in the field of Ramsey Theory. His collaborative spirit and unique approach to mathematics led to the development of numerous concepts that have become foundational in various mathematical disciplines.
Pigeonhole Principle: The pigeonhole principle is a simple yet powerful concept in combinatorics stating that if you have more items than containers to put them in, at least one container must hold more than one item. This principle underpins many results in various fields, illustrating that certain configurations or outcomes are unavoidable when the number of elements exceeds available categories.
Subadditivity: Subadditivity is a property of a function or a quantity that indicates the whole is less than or equal to the sum of its parts. In the context of Van der Waerden numbers, this concept helps in understanding how the formation of monochromatic arithmetic progressions in colored sets can be analyzed by breaking down larger sets into smaller ones, allowing for a more manageable investigation of their properties.
Van der Waerden number: A van der Waerden number, denoted as $W(k, r)$, is the smallest integer such that any coloring of the integers from 1 to $W(k, r)$ using $r$ colors guarantees that there exists a monochromatic arithmetic progression of length $k$. This concept connects deeply to the intersection of combinatorics and number theory, illustrating how order can emerge from chaos in colored sequences.
Van der Waerden's Theorem: Van der Waerden's Theorem states that for any given positive integers $r$ and $k$, there exists a minimum integer $N$ such that if the integers $1, 2, \, \ldots, \, N$ are colored with $r$ different colors, there will always be a monochromatic arithmetic progression of length $k$. This theorem connects to various areas of mathematics by illustrating how partitioning sets can lead to guaranteed structures within them.
W(3, 2): The term w(3, 2) refers to a specific Ramsey number that represents the smallest integer n such that any coloring of the edges of a complete graph on n vertices using two colors will guarantee the existence of a monochromatic triangle (a complete subgraph of three vertices) or a monochromatic edge (a complete subgraph of two vertices). Understanding w(3, 2) is key in studying Van der Waerden numbers, as it highlights the interplay between combinatorial structures and colorings, emphasizing how certain arrangements always lead to specific outcomes.
W(4, 2): The term w(4, 2) represents a specific value in Ramsey Theory known as a Ramsey number, which indicates the minimum number of vertices required to guarantee a complete graph can be colored with four colors such that no monochromatic complete subgraph of size two exists. It reflects the deep connection between combinatorics and graph theory, illustrating how structure and colorings interact in mathematical settings.
W(5, 2): The term w(5, 2) refers to a specific Van der Waerden number that represents the smallest integer n such that in any coloring of the integers from 1 to n with two colors, there exists a monochromatic arithmetic progression of length 5. This concept connects to Ramsey Theory and highlights the relationship between combinatorial coloring and arithmetic sequences.
W(k, r): The notation w(k, r) represents the smallest integer n such that any k-coloring of the integers {1, 2, ..., n} contains a monochromatic arithmetic progression of length r. This concept is crucial in Ramsey Theory as it connects coloring problems with structured combinatorial arrangements. It highlights how order and organization emerge from seemingly chaotic conditions, demonstrating the intricate balance between creativity and structure in mathematics.
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