Van der Waerden's theorem on arithmetic progressions -- a survey of some historical and modern developments

Abstract

Van der Waerden's theorem, published in Nieuw. Arch. Wisk. 15 (1927), acted as a catalyst for major further developments in Ramsey theory. In this survey, we delve into the legacy of this mathematical gem, tracing its historical origin, exploring its wide-ranging connections across research areas, and highlighting some of the important results it has inspired.

Figures (5)

• Figure 1: This diagram depicts the implications between the equivalent forms of van der Waerden's theorem considered in this survey. An arrow in the diagram with a label next to it corresponds to an implication for which we provide a proof in \ref{['sec_proof_of_equivalentcies_of_vdW']}, and the label specifies the key technical result underpinning this proof. Unlabeled arrows indicate immediate implications that do not require a proof. Below is a table that clarifies the abbreviations used in this diagram.
• Figure 2: This implication diagram provides a quick overview of the equivalent forms of Rado's theorem discussed in this subsection.
• Figure 3: This diagram summarizes the implications proved among the formulations of the IP van der Waerden theorem presented in this section. It also shows how the Gallai-Witt theorem and the affine spaces theorem are derived as corollaries from the IP van der Waerden theorem.
• Figure 4: This figure provides a synopsis of the content of this section. We present three equivalent forms of the Hales-Jewett theorem, introduce a new extension of the Hales-Jewett theorem, and prove that the Hales-Jewett theorem implies the IP van der Waerden theorem.
• Figure 5: The above diagram describes the connections between the polynomial extensions of van der Waerden's theorem that we deal with in this section.

Theorems & Definitions (121)

• proof : Proof of the \ref{['nam_brauer-schur']} assuming van der Waerden's theorem
• proof : Proof of \ref{['conj_schur_quadratic_residues']}
• Theorem 2.1: Finitary van der Waerden's theorem (Fin.vdW)
• Theorem 2.2: Infinitary van der Waerden's theorem (Inf.vdW)
• Theorem 2.3: van der Waerden's theorem for syndetic sets (vdW-syn)
• Theorem 2.4: van der Waerden's theorem for piecewise syndetic sets (vdW-pws)
• Theorem 2.5: van der Waerden's thm. for piecewise syndetic sets -- amplified version (vdW-pws$^+$)
• Theorem 2.6: van der Waerden's theorem for multiplicatively syndetic sets (vdW-mult.syn)
• Theorem 2.7: van der Waerden's thm. for multiplicatively piecewise syndetic sets (vdW-mult.pws)
• Theorem 2.8: Topological Multiple Recurrence Theorem -- set recurrence version (TMR-sets)