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.

• The paper traces van der Waerden’s theorem from quadratic-residue problems to modern partition Ramsey theory, explaining its relationship with Schur’s conjecture and the Brauer–Schur theorem.
• Van der Waerden’s theorem is shown equivalent to 11 combinatorial, topological, and ultrafilter-algebraic formulations, supported by compactness, recurrence, correspondence, and semigroup methods.
• The survey presents new and unified routes to Rado, IP, Hales–Jewett, and polynomial extensions while identifying open problems for IP configurations in amenable groups and density polynomial Hales–Jewett theory.

Historical origins: from quadratic residues to van der Waerden

The survey by Bergelson and Richter traces the theorem to a concrete number-theoretic problem. Jacobsthal showed in his 1906 thesis that for all sufficiently large primes $p$ there exist four consecutive quadratic residues and four consecutive quadratic nonresidues modulo $p$, and Schur conjectured the corresponding statement for arbitrary lengths $k$. Schur recognized that the residue case would follow from a purely combinatorial assertion: for every $k$ and sufficiently large $N$, any two-coloring of $\{1,\ldots,N\}$ contains a monochromatic $k$-term arithmetic progression together with its common difference. Van der Waerden proved the weaker statement without the common difference in 1927 — formulated, according to his own account, as a conjecture of Baudet, though no written record from Baudet exists and the question of independent conception versus transmission of Schur's conjecture remains historically unresolved (the authors cite Soifer's investigation supporting independence).

The gap between van der Waerden's theorem and Schur's needs was closed within days. Brauer and Schur jointly established what the authors call the Brauer-Schur theorem: for every $r,k$, any $r$-coloring of $\{1,\ldots,N\}$ (with $p$0 large) yields a monochromatic $p$1-term progression together with its difference. The proof is a short induction on $p$2: a monochromatic progression of length $p$3 is found via van der Waerden's theorem, and either one of $p$4 shares its color or the induction hypothesis applies to the remaining colors. From this, Schur's original conjecture on consecutive residues follows by dividing a monochromatic progression-with-difference by $p$5; the nonresidue case requires an additional argument exploiting the primality of the smallest quadratic nonresidue.

Eleven equivalent formulations

A central contribution of the survey is a systematic presentation of eleven equivalent forms of van der Waerden's theorem, organized into combinatorial, topological, and algebraic categories, with full proofs of the equivalences arranged in an implication diagram. The combinatorial versions include:

• the finitary form with van der Waerden numbers $p$6;
• the infinitary form;
• versions for syndetic sets, piecewise syndetic sets, and an "amplified" version asserting that every piecewise syndetic set contains a piecewise syndetic set of starters sharing a common difference;
• multiplicative analogues for multiplicatively syndetic and multiplicatively piecewise syndetic sets.

The topological versions are Furstenberg–Weiss style multiple recurrence statements for minimal systems: a set-recurrence form ($p$7) and a point-recurrence form asserting that $p$8-recurrent points form a residual set. The algebraic versions live on $p$9, the Stone–Čech compactification: there exist ultrafilters on the semigroup $k$0 of arithmetic-progression tuples whose coordinate projections coincide, and this can be strengthened so that the common projection is any prescribed minimal ultrafilter.

The proofs of equivalence introduce several reusable tools: the compactness principle (via Tychonoff), Brown's lemma and its strengthening to partition regularity of piecewise syndeticity, intersectivity lemmas for (multiplicatively) piecewise syndetic sets, a topological correspondence principle relating finite colorings to minimal systems built from shift spaces, and Ellis's lemma on idempotents in right-topological semigroups. Notably, the correspondence principle for piecewise syndetic sets yields a quick route from topological multiple recurrence to the amplified piecewise-syndetic version.

Three proofs of the theorem

The survey presents three proofs, each highlighting a different technique. The combinatorial proof departs from the classical double induction (on $k$1 and $k$2): instead, a single induction on $k$3 is driven by the intersectivity lemma for piecewise syndetic sets, producing nested monochromatic piecewise syndetic sets $k$4 with differences $k$5, and a pigeonhole argument over the finitely many colors closes a $k$6-term progression. This proof transparently exhibits why differences of the form $k$7 arise — the observation motivating IP extensions.

The topological proof rests on a topological analogue of van der Corput's difference theorem: if all finite "difference families" of a sequence family are families of multiple topological recurrence, then so is the original family. Applied to $k$8 with induction on $k$9, this yields the set-recurrence theorem for minimal systems. The authors emphasize that this approach upgrades readily to polynomial sequences.

The algebraic proof, developed in part through private communication with Neil Hindman, works in $k$0 for a carefully chosen subsemigroup $k$1. The key distinction from prior renditions is taking closures inside $k$2 rather than $k$3, then combining existence of idempotents in minimal left ideals with the fact that an idempotent in a minimal left ideal acts as a right identity on it, forcing all coordinate projections of a suitable ultrafilter onto a prescribed minimal ultrafilter.

Outgrowths: Rado, IP, Hales–Jewett, and polynomial extensions

Rado's theorem characterizes partition regularity of homogeneous linear systems $k$4 via the columns condition. The survey gives new equivalent formulations stating that every multiplicatively syndetic (resp. multiplicatively piecewise syndetic) set contains solutions exactly when the columns condition holds — a structural refinement of a different flavor than central sets or $k$5-sets, since multiplicatively piecewise syndetic sets may have zero additive Banach density. The sufficiency proof proceeds by PET-style induction on the level of the columns condition, using a reduction lemma and the multiplicative van der Waerden theorem; necessity uses Rado's $k$6-colorings based on residues modulo primes.

The IP van der Waerden theorem replaces linear progressions by configurations $k$7 along IP-sequences, valid in arbitrary abelian groups. The survey proves three equivalent forms (combinatorial, simplex, topological) and derives two corollaries: the Gallai–Witt theorem on monochromatic homothetic copies of finite sets in $k$8, and the affine spaces theorem guaranteeing monochromatic affine subspaces of any dimension in finite colorings of vector spaces over finite fields.

For the Hales–Jewett theorem, the authors prove a new generalization — a lifting theorem for arbitrary semigroups: given a subsemigroup $k$9, a two-sided ideal $N$0, and homomorphisms $N$1, every minimal ultrafilter on $N$2 arises as the common projection of a minimal ultrafilter on $N$3. Instantiated with variable words over $N$4, this yields the Hales–Jewett theorem, and the authors show Hales–Jewett implies the IP van der Waerden theorem.

Finally, the polynomial van der Waerden theorem (Bergelson–Leibman) is proved in its one-dimensional topological form via PET-induction on weight vectors, with the topological van der Corput difference theorem supplying the complexity-reduction step; the quadratic case ($N$5 monochromatic) is treated separately as a warm-up. The section closes with the multidimensional polynomial extension and the polynomial Hales–Jewett theorem, the most general coloring result discussed.

Open problems

Two conjectures occupy the final section. First, an IP van der Waerden theorem for amenable groups: for finite partitions of $N$6, some cell should contain simplices whose return-time set is an IP$N$7-set. Partial results exist — the case $N$8 was recently settled, nilpotent groups follow from Bergelson–Leibman–Moreira, but for $N$9 and non-nilpotent groups the problem is open; the failure for infinitely generated free groups indicates amenability is the natural setting. Second, the thirty-year-old density polynomial Hales–Jewett conjecture of Bergelson, which would substantially strengthen existing polynomial Szemerédi-type theorems; it remains open.

Conclusion

This survey positions van der Waerden's theorem as a hub of partition Ramsey theory, demonstrating through complete proofs that its combinatorial, topological-dynamical, and ultrafilter-algebraic incarnations are genuinely equivalent rather than merely parallel. The new contributions — the intersectivity-driven combinatorial proof, the van der Corput-based topological proof extending to polynomials, the Hindman-inspired algebraic proof, the semigroup lifting theorem yielding Hales–Jewett, and the multiplicative-largeness approach to Rado's theorem — illustrate how each perspective generates distinct generalizations, while the two closing conjectures delineate precisely where the abelian, coloring-level theory ends and open territory begins.