4.2 Van der Waerden's theorem

Van der Waerden's theorem is a game-changer in arithmetic progressions. It says that no matter how you slice up the numbers, you'll always find patterns. This idea has far-reaching effects in math and beyond.

The proof is a bit tricky, using clever tricks like the pigeonhole principle. It doesn't give us exact numbers, but it shows these patterns exist. This opens up a whole world of questions about number patterns.

Significance of Van der Waerden's theorem

Statement and implications

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• Van der Waerden's theorem states for any positive integers $r$ and $k$, there exists a positive integer $N$ such that if the set ${1, 2, ..., N}$ is partitioned into $r$ subsets, then at least one of the subsets contains an arithmetic progression of length $k$
• Establishes the existence of arithmetic progressions in any finite coloring of the positive integers, regardless of the number of colors used or the length of the progression desired
• Fundamental result in Ramsey theory studies the conditions under which certain patterns must appear in large structures
• Has applications in various areas of mathematics (number theory, combinatorics, computer science)

Proof characteristics

• The proof of Van der Waerden's theorem is non-constructive
  • Does not provide an explicit value for $N$ given $r$ and $k$
  • Establishes its existence using the pigeonhole principle and mathematical induction

Proof of Van der Waerden's theorem

Defining the Van der Waerden number

• Define the Van der Waerden number $W(r, k)$ as the smallest positive integer $N$ such that any $r$-coloring of ${1, 2, ..., N}$ contains a monochromatic arithmetic progression of length $k$
• Prove the base case: For $k = 1$, $W(r, 1) = 1$ for any $r$, as any single integer forms a trivial arithmetic progression of length 1

Inductive step

• Assume that $W(r, k)$ exists for all $r$ and $k$ up to some fixed value of $k$
• Show that $W(r, k+1)$ exists by considering an $r$-coloring of ${1, 2, ..., W(r^{W_k}, k)}$, where $W_k = W(r, k)$
• Use the pigeonhole principle to argue in the $r$-coloring of ${1, 2, ..., W(r^{W_k}, k)}$, there must be a monochromatic subset of size at least $W_k$
• Apply the inductive hypothesis to the monochromatic subset, showing it must contain a monochromatic arithmetic progression of length $k$
• Demonstrate the arithmetic progression of length $k$, along with the common difference between its terms, forms a monochromatic arithmetic progression of length $k+1$ in the original $r$-coloring
• Conclude that $W(r, k+1)$ exists and is at most $W(r^{W_k}, k)$, completing the inductive step and proving the theorem

Applications of Van der Waerden's theorem

Additive combinatorics

• Prove the existence of arbitrarily long arithmetic progressions in the set of prime numbers
• Prove Szemerédi's theorem states any set of integers with positive upper density contains arbitrarily long arithmetic progressions

Ramsey theory

• Establish lower bounds on the Ramsey number $R(k, k)$, the smallest positive integer $N$ such that any 2-coloring of the edges of the complete graph on $N$ vertices contains a monochromatic complete subgraph on $k$ vertices
• Prove the Hales-Jewett theorem, a generalization of Van der Waerden's theorem to higher dimensions

Connections to other theorems and conjectures

• Investigate the relationship between Van der Waerden's theorem and the Green-Tao theorem states the set of prime numbers contains arbitrarily long arithmetic progressions
• Explore the connection between Van der Waerden's theorem and the Erdős-Turán conjecture concerns the existence of arithmetic progressions in sets with positive density