Three-color van der Waerden numbers grow super-exponentially

Abstract: For $k$ sufficiently large, we show that there is a three-coloring of the first $2^{k (\log^* k)/4}$ positive integers without any monochromatic $k$-term arithmetic progressions. Thus, the three-color van der Waerden number $w(k;3)$ grows faster than any exponential in $k$. We further prove a new lower bound on multicolor van der Waerden numbers which resolves a problem of Erdős and Graham on canonical van der Waerden numbers.

• The paper proves that for all sufficiently large k, the three-color van der Waerden number satisfies w(k;3) > 2^{k(log* k)/4}, resolving the question of super-exponential growth for three colors.
• The authors combine dense arithmetic-progression-free sets, Lovász local lemma colorings, cyclic hitting sets, and random shifted products to iterate colorings while keeping the number of colors fixed.
• The results extend to many colors, yielding w(k;r) ≥ r^{(1−o(1))k log k} for sufficiently large r and resolving the Erdős–Graham problem with H(k) ≥ k^{(1−o(1))k log k}, while the two-color case remains open.

Overview and main results

This paper, by Jacob Fox and Zach Hunter (2606.02541), establishes that the three-color van der Waerden number $w(k;3)$ grows faster than any exponential function of $k$. Specifically, for all sufficiently large $k$,

$w(k;3) > 2^{k(\log^* k)/4},$

where $\log^* k$ denotes the iterated logarithm. This resolves a long-standing question in Ramsey theory: Erdős had offered $500 to prove or disprove super-exponential growth of$w(k;2)$, guessing it holds but expecting a difficult proof. The paper shows the answer is affirmative already at three colors (and hence for all$r \geq 3$), thereby refuting the conjecture appearing in several sources that$\lim_{k\to\infty} w(k;r)^{1/k} = r$for every fixed$r$. The authors note that prior work of Hunter [2606.02541's cited reference] had already disproved this conjecture for$r \geq 5$k$0(3<sup>{r/3})<sup>{(1-o(1))k}$k$1r = 3$.

A second main theorem gives a lower bound in the many-color regime: for each $k$2 there is $k$3 such that if $k$4 and $k$5, then

$k$6

As a corollary, the canonical van der Waerden number $k$7 — the least $k$8 such that every coloring of $k$9 contains a monochromatic or rainbow $k$0-term arithmetic progression — satisfies

$k$1

which resolves an open problem of Erdős and Graham asking whether $k$2. The authors point out that independent recent work of Bae obtained the weaker bound $k$3, which also settles the problem.

Dense sets without long arithmetic progressions

The construction begins from the classical Erdős–Turán digit-restriction set: for a prime $k$4, the integers whose base-$k$5 expansion avoids the digit $k$6 contain no $k$7-term arithmetic progression and occupy a fraction $k$8 of $k$9. Consequently, when $w(k;3) > 2^{k(\log^* k)/4},$0, there are subsets of $w(k;3) > 2^{k(\log^* k)/4},$1 of density $w(k;3) > 2^{k(\log^* k)/4},$2 with no $w(k;3) > 2^{k(\log^* k)/4},$3-term arithmetic progression. The authors transfer this to cyclic groups via a product argument over prime-power factors, obtaining a $w(k;3) > 2^{k(\log^* k)/4},$4-AP-free subset of $w(k;3) > 2^{k(\log^* k)/4},$5 of density at least $w(k;3) > 2^{k(\log^* k)/4},$6 when $w(k;3) > 2^{k(\log^* k)/4},$7 with primes $w(k;3) > 2^{k(\log^* k)/4},$8. They also record the monotone parameter $w(k;3) > 2^{k(\log^* k)/4},$9, which satisfies $\log^* k$0 for $\log^* k$1.

A technical complication is that arithmetic progressions in $\log^* k$2 may wrap around the cut between $\log^* k$3 and $\log^* k$4. The paper handles this with a Fourier-inspired but purely combinatorial lemma: for prime $\log^* k$5, there is a subset $\log^* k$6 of density at most $\log^* k$7 meeting every wrapping $\log^* k$8-AP in at least an $\log^* k$9-fraction. The proof takes $500 to prove or disprove super-exponential growth of$0, where $500 to prove or disprove super-exponential growth of$1 is a balanced interval of length about $500 to prove or disprove super-exponential growth of$2 and $500 to prove or disprove super-exponential growth of$3; Dirichlet approximation ensures any progression either has large common difference (so is well-distributed over rotations by a counting lemma on balanced intervals) or has small structured difference (so intersects $500 to prove or disprove super-exponential growth of$4 directly).

The key structural input is Lemma "cyclicdense": for $500 to prove or disprove super-exponential growth of$5 a product of $500 to prove or disprove super-exponential growth of$6 distinct primes in $500 to prove or disprove super-exponential growth of$7 with $500 to prove or disprove super-exponential growth of$8, there is a subset $500 to prove or disprove super-exponential growth of$9 of density at most $$, guessing it holds but expecting a difficult proof. The paper shows the answer is affirmative already at three colors (and hence for all$$0 containing at least a $$, guessing it holds but expecting a difficult proof. The paper shows the answer is affirmative already at three colors (and hence for all$$1-fraction of every $$, guessing it holds but expecting a difficult proof. The paper shows the answer is affirmative already at three colors (and hence for all$$2-AP, where one may take $$, guessing it holds but expecting a difficult proof. The paper shows the answer is affirmative already at three colors (and hence for all$$3. The interval version is proved by a Lovász-local-lemma-style random construction: for each prime $$, guessing it holds but expecting a difficult proof. The paper shows the answer is affirmative already at three colors (and hence for all$$4, choose $$, guessing it holds but expecting a difficult proof. The paper shows the answer is affirmative already at three colors (and hence for all$$5 random residue classes mod $$, guessing it holds but expecting a difficult proof. The paper shows the answer is affirmative already at three colors (and hence for all$$6; a union bound over progressions and deficient subsets shows positive probability that $$, guessing it holds but expecting a difficult proof. The paper shows the answer is affirmative already at three colors (and hence for all$$7 has density at most $$, guessing it holds but expecting a difficult proof. The paper shows the answer is affirmative already at three colors (and hence for all$$8 yet meets every $$, guessing it holds but expecting a difficult proof. The paper shows the answer is affirmative already at three colors (and hence for all$$9-AP in at least $$), thereby refuting the conjecture appearing in several sources that$$0 fraction. The cyclic version follows by combining the interval construction with the wrapping-AP hitting set via the Chinese remainder theorem and a product lemma. Note the dependence of $$), thereby refuting the conjecture appearing in several sources that$$1 on $$), thereby refuting the conjecture appearing in several sources that$$2 is doubly exponential in $$), thereby refuting the conjecture appearing in several sources that$$3; this loss governs how many iterations the final construction tolerates.

Local lemma colorings and random shifted products

Two further ingredients feed the iteration. First, a consequence of the symmetric Lovász local lemma shows that if $$), thereby refuting the conjecture appearing in several sources that$$4 is an abelian group of order $$), thereby refuting the conjecture appearing in several sources that$$5 with no prime factor below $$), thereby refuting the conjecture appearing in several sources that$$6, then elements can be assigned $$), thereby refuting the conjecture appearing in several sources that$$7-subsets of an $$), thereby refuting the conjecture appearing in several sources that$$8-color set so that no color appears on more than $$), thereby refuting the conjecture appearing in several sources that$$9 vertices of any $for every fixed$0-AP. In particular, for $for every fixed$1, $for every fixed$2, any abelian group of order up to roughly $for every fixed$3 admits a two-coloring in which each $for every fixed$4-AP has at most $for every fixed$5 elements of either color.

Second, the paper develops two variants of the random shifted product construction. Given an $for every fixed$6-coloring of $for every fixed$7 that is far from monochromatic on every $for every fixed$8-AP and has a dense color class, and similarly for $for every fixed$9, one colors fibers of the projection $$. The authors note that prior work of Hunter [2606.02541's cited reference] had already disproved this conjecture for$$0 by independent random translates of the $$. The authors note that prior work of Hunter [2606.02541's cited reference] had already disproved this conjecture for$$1-coloring, with colors permuted according to the $$. The authors note that prior work of Hunter [2606.02541's cited reference] had already disproved this conjecture for$$2-color. A union bound over the at most $$. The authors note that prior work of Hunter [2606.02541's cited reference] had already disproved this conjecture for$$3 progressions yields an output coloring of $$. The authors note that prior work of Hunter [2606.02541's cited reference] had already disproved this conjecture for$$4 with the same qualitative properties, provided $$. The authors note that prior work of Hunter [2606.02541's cited reference] had already disproved this conjecture for$$5. A set-coloring-assisted variant instead consumes a coloring of $$. The authors note that prior work of Hunter [2606.02541's cited reference] had already disproved this conjecture for$$6 by $$. The authors note that prior work of Hunter [2606.02541's cited reference] had already disproved this conjecture for$$7-subsets of $$. The authors note that prior work of Hunter [2606.02541's cited reference] had already disproved this conjecture for$$8 and produces an ordinary $$. The authors note that prior work of Hunter [2606.02541's cited reference] had already disproved this conjecture for$$9-coloring of the product without monochromatic $k$00-APs when $k$01. Compared to the earlier product constructions in Hunter's work, these versions keep the number of colors fixed through the iteration, which is what makes the three-color result possible. The authors remark that replacing the union bound by the local lemma would improve the conditions by a factor involving $k$02, but only yield minor gains in applications.

Proof architecture of the main bound

The three-color lower bound proceeds by induction over $k$03 steps. Let $k$04 be distinct primes near $k$05 and $k$06. At step $k$07, the complement of the dense-hitting-set construction supplies a subset $k$08 of density at least $k$09 meeting every $k$10-AP in at most a $k$11 fraction, with $k$12 and $k$13; combining this with the local-lemma two-coloring gives a three-coloring of $k$14 with a very dense third color. The shifted product lemma then merges this with the coloring of $k$15: the group size grows by a factor exponential in $k$16 (each $k$17), while the density parameter degrades via a double-exponential recurrence $k$18. Since $k$19, the parameters remain valid for all $k$20 steps, producing a three-coloring of $k$21 with $k$22 and no monochromatic $k$23-AP. The exponent carries a factor of $k$24 arising from constants in the density bookkeeping; tightening these constants would improve the base of the growth but not its super-exponential character.

For the many-color regime, Theorem "masterpower" runs a similar recursion with $k$25 prime factors near $k$26, adding $k$27 colors per step via Lemma "addscolors general" and the set-coloring-assisted product lemma. Choosing $k$28 and optimizing yields Corollary "lastcor": for $k$29, $k$30 with $k$31, which implies Theorem "powervdw". The authors state plainly that improving the exponent beyond $k$32 is bottlenecked by the absence of a construction of a $k$33-AP-free set of density $k$34 inside $k$35 when $k$36; any such construction would immediately strengthen the bound through their master theorem. For very large $k$37 relative to $k$38, Rankin-type constructions give better bounds, and combined with the Kelley–Meka theorem one has $k$39.

Limitations and open questions

Several caveats qualify the results. The super-exponential bound applies only for $k$40; whether $k$41 itself grows super-exponentially — Erdős's original conjecture, which he suspected true — remains open, as does the asymptotic behavior of $k$42. The constant $k$43 in the exponent of Theorem "super-exponential" is an artifact of the analysis rather than intrinsic, and the authors' own remark indicates the union-bound conditions in the product lemmas could be sharpened via the local lemma, though with minor effect. The many-color bound requires $k$44, i.e., $k$45 polylogarithmic in $k$46 or larger; the intermediate regime and the exact dependence of $k$47 on $k$48 are left open, contingent on the density-construction bottleneck described above. Finally, the paper notes that parts of the framework are developed in greater generality than needed here, with applications to other arithmetic Ramsey numbers deferred to a forthcoming companion work, and the authors suggest the techniques should yield improved lower bounds for concrete small van der Waerden numbers with at least three colors.

Conclusion

This paper proves that $k$49 for all sufficiently large $k$50, establishing super-exponential growth of multicolor van der Waerden numbers and refuting the conjecture that $k$51. The proof combines Erdős–Turán-type dense AP-free sets, a robust sparse hitting-set construction valid in cyclic groups up to tower-height constraints, local-lemma colorings, and efficient random shifted product iterations that preserve the number of colors. The same machinery yields $k$52 once $k$53 and resolves the Erdős–Graham problem on canonical van der Waerden numbers via $k$54. The central open questions left by the paper concern the two-color case, the removal of the density bottleneck limiting the many-color exponent, and the optimal constants in the super-exponential rate.