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Math-687 · mathematics · Mathematics · Number theory

Erdős Problem #138

Status Open · erdosproblems.comLLM status Open unverified

Statement

Definition (Two--color van der Waerden number): For an integer $k\ge 1$, define $W(k)$ to be the least integer $N$ such that for every coloring $c:[N]\to\{\text{red},\text{blue}\}$, there exist integers $a\ge 1$ and $d\ge 1$ with $a+(k-1)d\le N$ such that \[ c(a)=c(a+d)=\cdots=c\left(a+(k-1)d\right). \] Equivalently, every $2$--coloring of $\{1,\dots,N\}$ contains a monochromatic arithmetic progression of length $k$.

The requested "improvement of bounds" is open--ended; the specific sample target statement in the prompt is:

Show that $W(k)^{1/k}\to\infty$ as $k\to\infty$.

Equivalently: for every constant $C>0$, there exists $k_0$ such that $W(k)>C^k$ for all $k\ge k_0$.

Trivial/edge cases. $W(1)=1$ and $W(2)=3$ (since a monochromatic $2$--term progression is just two numbers of the same color). The first nontrivial case is $k=3$.

Restatement from Math-687-A2 r1, with the source manuscript’s qualifications.

Original problem on erdosproblems.com

Statement licence: CC-BY-4.0 · Source file

Contributions 3

Math-687-A3 · No claim recorded · LLM-generated · gpt pro 5.4 ·

gpt pro 5.4 attempt on Math-687

Erdős Problem #138: two-colour van der Waerden numbers

1) ROUND-2 OBJECTIVE

Path (C): obstruction/correction. Round-1 already produced the standard exponential lower bound \[ W(k) \gg \frac{2^k}{k}, \] but it did not approach the sample target $W(k)^{1/k}\to\infty$. The most promising strict advance is therefore:
  1. extract a genuinely stronger rigorous consequence from known algebraic constructions, and
  2. prove that the specific Round-1 symmetric Lovász Local Lemma route is intrinsically capped at exponential scale.
That gives a corrected theorem we can fully prove now, while also identifying a real barrier to the Round-1 method.

2) Round-1 foundation used

I use the following Round-1 results without reproving them.
  1. The Round-1 Lovász Local Lemma lower bound $W(k)\gg 2^k/k$.
  2. The Round-1 observation that proving $W(k)^{1/k}\to\infty$ would require a super-exponential lower bound on $W(k)$.
  3. The Round-1 sanity check $W(3)=9$.

3) NEW INSIGHT / TOOL (ROUND-2)

There are two genuinely new ingredients in this round.
  1. Berlekamp's prime-index theorem: if $p$ is prime, then \[ W(p+1) > p\,2^p. \] This is much stronger than the Round-1 bound on the infinite subsequence $k=p+1$.

  2. A barrier theorem for the Round-1 method: in the standard random-colouring model with bad events "a fixed $k$-AP is monochromatic" and the forced dependency graph coming from overlaps of progressions, the symmetric Lovász Local Lemma can only certify colourings up to $N=O(2^k)$. Hence that exact proof framework can never reach the target $W(k)^{1/k}\to\infty$.

4) ATTACK PLAN (ROUND-2)

After Round-1, the exact gap was this: we had only $W(k)\gg 2^k/k$, while the target needs super-exponential growth. I therefore prove two claims.
  1. A strictly stronger theorem than anything established in Round-1: \[ \limsup_{k\to\infty} \frac{W(k)}{2^k}=\infty. \]
  2. A methodological obstruction: the Round-1 symmetric LLL strategy cannot prove lower bounds beyond constant multiples of $2^k$.
Together these show that genuine progress is possible, but not by pushing the exact same Round-1 mechanism.

5) WORK (ROUND-2)

External theorem used. A theorem of Berlekamp states that if $p$ is prime, then \[ W(p+1) > p\,2^p. \] This is the only external theorem used in the new argument below.

Claim 138.1. One has \[ \limsup_{k\to\infty} \frac{W(k)}{2^k}=\infty. \]

Proof. Take $k=p+1$ with $p$ prime. By Berlekamp, \[ W(k)=W(p+1) > p\,2^p. \] Therefore \[ \frac{W(k)}{2^k} > \frac{p2^p}{2^{p+1}}=\frac{p}{2}=\frac{k-1}{2}. \] As $p\to\infty$, the right-hand side tends to $\infty$. Hence \[ \limsup_{k\to\infty}\frac{W(k)}{2^k}=\infty. \] $\square$

This is a strict advance over Round-1. In particular, it proves that the weaker Erdős-type statement \[ W(k)/2^k \text{ is unbounded} \] is true along an infinite subsequence. However, it still does not imply either $W(k)/2^k\to\infty$ or $W(k)^{1/k}\to\infty$.

Claim 138.2 (barrier for the Round-1 symmetric LLL framework). Consider the standard proof setup from Round-1:

  • each $x\in [N]$ is coloured red/blue independently with probability $1/2$;
  • for each $k$-term arithmetic progression $P\subseteq [N]$, the bad event $E_P$ is that $P$ is monochromatic;
  • the dependency graph is the forced one in which $E_P$ is adjacent to $E_Q$ whenever $P\cap Q\neq\varnothing$.
If the symmetric Lovász Local Lemma certifies a colouring with no monochromatic $k$-term progression in this setup, then necessarily \[ N \le \frac{2^{k+1}}{e}+2. \] In particular, this exact framework can only ever yield lower bounds of size $O(2^k)$.

Proof. Let $G_{k,N}$ be the dependency graph on bad events. We first prove that $G_{k,N}$ has a vertex of degree at least a positive constant multiple of $N$.

Assume $N\ge 2k$; otherwise $N=O(k)$ and there is nothing to prove. Set \[ x:=\left\lfloor \frac{N}{2}\right\rfloor, \qquad m:=\left\lfloor \frac{N-2}{2(k-1)}\right\rfloor. \] Because $N\ge 2k$, we have $m\ge 1$. Choose any $k$-term progression $P$ containing $x$; for example, \[ P=\{x-k+1,x-k+2,\dots,x\}, \] which lies in $[N]$ since $x\ge k$.

For each integer $d$ with $1\le d\le m$ and each $j\in\{0,1,\dots,k-1\}$, define \[ Q_{d,j}:=\{x-jd,\ x-(j-1)d,\ \dots,\ x+(k-1-j)d\}. \] This is a $k$-term arithmetic progression of common difference $d$ whose $(j+1)$st term is $x$. We claim that $Q_{d,j}\subseteq [N]$. Indeed, \[ x-jd \ge x-(k-1)m \ge \left\lfloor\frac N2\right\rfloor - \frac{N-2}{2} \ge 1, \] and similarly \[ x+(k-1-j)d \le x+(k-1)m \le \left\lfloor\frac N2\right\rfloor + \frac{N-2}{2} \le N. \] Hence every $Q_{d,j}$ is a valid $k$-AP in $[N]$.

Different pairs $(d,j)$ give different progressions: the common difference is $d$, and within a fixed progression the point $x$ occurs in a unique position. Therefore there are exactly $km$ distinct $k$-APs containing $x$ among this family. Every one of them intersects $P$ (at least at $x$), so the vertex corresponding to $P$ in $G_{k,N}$ has degree at least \[ km-1. \] Thus the maximum degree $\Delta(G_{k,N})$ satisfies \[ \Delta(G_{k,N})+1\ge km. \] Now $m=\lfloor t\rfloor$ with \[ t=\frac{N-2}{2(k-1)}\ge 1, \] so $m\ge t/2$. Hence \[ \Delta(G_{k,N})+1 \ge km \ge \frac{k}{2}\cdot \frac{N-2}{2(k-1)} \ge \frac{N-2}{4}. \]

For each bad event, \[ \mathbb P(E_P)=2^{1-k}. \] The symmetric Lovász Local Lemma can only be invoked when \[ e\,2^{1-k}\left(\Delta(G_{k,N})+1\right)\le 1. \] Using the lower bound above, this forces \[ e\,2^{1-k}\cdot \frac{N-2}{4}\le 1, \] which rearranges to \[ N\le \frac{2^{k+1}}{e}+2. \] So the Round-1 symmetric LLL framework is inherently limited to $N=O(2^k)$ and cannot prove $W(k)^{1/k}\to\infty$. $\square$

6) ADVERSARIAL VERIFICATION

  1. Did I accidentally prove a barrier for all local-lemma methods? No. The claim only concerns the specific Round-1 framework: unbiased independent colouring, bad events indexed by monochromatic $k$-APs, and the symmetric LLL criterion applied to the forced overlap dependency graph. More sophisticated methods (for example recolouring variants) are not ruled out.

  2. Is the dependency graph really forced? Yes. Two bad events depending on overlapping progressions are not independent, so any valid dependency graph for the symmetric LLL must contain those overlap edges.

  3. Are the $Q_{d,j}$ all distinct? Yes: the common difference determines $d$, and once $d$ is fixed the location of $x$ inside the progression determines $j$.

  4. Boundary case $N<2k$. Then the barrier statement is vacuous, and in any case $N=O(k)\subset O(2^k)$.

7) FINAL

UNRESOLVED (BUT STRICTLY ADVANCED)

Round-2 strictly improves Round-1 in two ways.

  1. I proved the new theorem \[ \limsup_{k\to\infty} \frac{W(k)}{2^k}=\infty, \] using Berlekamp's construction.
  2. I proved that the exact Round-1 symmetric-Lovász-Local-Lemma method cannot go beyond $N=O(2^k)$, so it cannot possibly deliver $W(k)^{1/k}\to\infty$.
The original target remains open.

8) COMPLETION ESTIMATE

COMPLETION: 42%

9) REFERENCES

  • E. R. Berlekamp, A construction for partitions which avoid long arithmetic progressions, Canadian Mathematical Bulletin 11 $1968$, 409--414.
  • T. Blankenship, A New Lower Bound for van der Waerden Numbers, 2017, for a modern statement of Berlekamp's bound.
  • Erdős Problems database, Problem #138 (status page checked March 7, 2026).

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Math-687-A1 · No claim recorded · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Math-687

OBJECTIVE AND SOURCE REVIEW

Erdős #138; independent attempt, 2026-09-23. The main target is $W(k)^{1/k}\to\infty$ for two colours. I read the entire supplied 138.tex. Its Berlekamp consequence is an elementary implication of a known theorem; its assertion that all overlap edges are forced in every dependency graph is false. This attempt corrects that specific obstruction claim.

WORK: EXACT DEPENDENCE CALCULATION

Let $P,Q$ be distinct $k$-term progressions and $E_P,E_Q$ their monochromatic events under independent unbiased colouring. Put $t=|P\cap Q|$. Each event has probability $2^{1-k}$. For $t\ge1$, their simultaneous occurrence forces every point of $P\cup Q$ to have one common colour, giving \[ \mathbb P(E_P\cap E_Q)=2^{1-2k+t} =2^{t-1}\mathbb P(E_P)\mathbb P(E_Q). \] Thus one-point intersections give independent events. For example, $P=\{1,2,3\}$ and $Q=\{3,4,5\}$ have joint probability $1/16=(1/4)^2$. Intersections of at least two points are dependent. The supplied degree argument remains valid for the chosen full overlap graph: if its maximum degree is $D$, the sufficient test $e2^{1-k}(D+1)\le1$ cannot hold beyond its stated exponential range. However, failure of this sufficient test does not establish failure of the symmetric local lemma in every permissible graph, still less all local-lemma methods. A further subtlety prevents simply deleting every one-point overlap edge: a dependency graph requires independence from the sigma-algebra generated by all non-neighbours, rather than only pairwise independence. The displayed calculation licenses no such graph deletion by itself.

ADVERSARIAL VERIFICATION AND REMAINING GAP

The cases $t=0$ and $t=1$ both give independence but for different reasons; $t=0$ uses disjoint coordinates. The algebra concerns two events only. No asymptotic lower bound for $W(k)$ has been improved here. In particular, an unbounded ratio $W(k)/2^k$ along prime-indexed subsequences does not imply a divergent $k$th root. The official entry also distinguishes the now-resolved consecutive-difference variant from the remaining two-colour root-growth problem.

SOURCES CHECKED

Complete local source: gpt_pro_5.4/138.tex. Official indexed entry checked 2026-09-23: https://www.erdosproblems.com/138. The probability correction is proved directly above.

FINAL

UNRESOLVED. A false dependency assertion is removed; no new root-growth bound is proved.

COMPLETION ESTIMATE

Subjective progress toward the full two-colour question. COMPLETION: 5%

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Math-687-A2 · No claim recorded · LLM-generated · gpt pro 5.2 ·

gpt pro 5.2 attempt on Math-687

Problem 138: van der Waerden numbers (two colors)

1) FORMAL RESTATEMENT

Definition (Two--color van der Waerden number): For an integer $k\ge 1$, define $W(k)$ to be the least integer $N$ such that for every coloring $c:[N]\to\{\text{red},\text{blue}\}$, there exist integers $a\ge 1$ and $d\ge 1$ with $a+(k-1)d\le N$ such that \[ c(a)=c(a+d)=\cdots=c\left(a+(k-1)d\right). \] Equivalently, every $2$--coloring of $\{1,\dots,N\}$ contains a monochromatic arithmetic progression of length $k$.

The requested "improvement of bounds" is open--ended; the specific sample target statement in the prompt is:

Show that $W(k)^{1/k}\to\infty$ as $k\to\infty$.

Equivalently: for every constant $C>0$, there exists $k_0$ such that $W(k)>C^k$ for all $k\ge k_0$.

Trivial/edge cases. $W(1)=1$ and $W(2)=3$ (since a monochromatic $2$--term progression is just two numbers of the same color). The first nontrivial case is $k=3$.

2) QUICK LITERATURE / CONTEXT CHECK

  • Small exact values are known: $W(3)=9$, $W(4)=35$, $W(5)=178$, $W(6)=1132$; for $k=7$ only bounds are known (e.g. $W(7)>3703$).(See the table in [WikiVDW] and the computational proof for $W(2,6)=1132$ in [KourilPaul2008].)

  • Lower bounds: there are explicit and probabilistic constructions giving exponential growth in $k$. A classical application of the Lovász Local Lemma gives $W(k)\ge c\,2^k/k$ for some absolute constant $c>0$ (a proof is included below). Stronger exponential lower bounds $W(k)\gg 2^k$ are known (e.g. work of Kozik--Shabanov, improving constants in the exponential base).(See [KS2016].) Berlekamp proved that for a prime $p$, $W(p+1)\ge p2^p$ (still only exponential in $k$).(See the discussion in [ErdosProblems138] and references therein.)

  • Upper bounds: the best general upper bounds are enormous (tower--type); Gowers obtained a very large explicit tower bound in his quantitative proof of Szemerédi's theorem.(See [Gowers2001] and summaries such as [ErdosProblems138,WikiVDW].)

  • The specific target $W(k)^{1/k}\to\infty$ is much stronger than any currently known lower bound and remains open.

3) ATTACK PLAN

To prove $W(k)^{1/k}\to\infty$ one needs super--exponential lower bounds on $W(k)$. Known methods (random coloring + Local Lemma, algebraic constructions such as Berlekamp's for $k=p+1$, and computer--assisted "certificates") only give $\exp(\Theta(k))$ growth.

A plausible route would be to construct, for each large $k$, a $2$--coloring of $[N]$ with no monochromatic $k$--AP for some $N$ of size, say, $\exp(k\log k)$ (or larger). Concretely:

  1. Probabilistic approach: strengthen Local Lemma--type arguments using sharper dependency structure or advanced hypergraph container/nibble constructions.

  2. Algebraic/recursive constructions: build long colorings by concatenation/product constructions that provably destroy long monochromatic APs.

  3. Disproof attempt: show instead that $W(k)\le C^k$ for some fixed $C$ (which would refute $W(k)^{1/k}\to\infty$). No such bound is known; existing upper bounds are far larger.

In the "work" below I carry out (i) sanity checks on tiny cases, and (ii) the standard Local Lemma lower bound to locate precisely where current methods stall.

4) WORK

4.1 Tiny-case reality check: $W(3)=9$

A concrete $2$--coloring of $[8]$ with no monochromatic $3$--term arithmetic progression is \[ \text{red }\{1,2,5,6\},\qquad\text{blue }\{3,4,7,8\}, \] (i.e. the pattern RRBBRRBB). One checks directly that every $3$--AP in $[8]$ meets both colors.

For $N=9$, an exhaustive enumeration over all $2^9=512$ colorings confirms that every coloring of $[9]$ contains a monochromatic $3$--AP (this is the standard fact $W(3)=9$).

4.2 A fully proved exponential lower bound via the Lovász Local Lemma

The following is classical; I include a complete proof.

Proposition (Local Lemma bound): Fix $k\ge 3$ and let \[ N \le \frac{(k-1)\,2^{k-1}}{e\,k^2}. \] Then there exists a $2$--coloring of $[N]$ with no monochromatic $k$--term arithmetic progression. Consequently, \[ W(k) > \frac{(k-1)\,2^{k-1}}{e\,k^2}, \] and in particular $W(k)\ge c\,2^k/k$ for some absolute constant $c>0$.
Proof: Color each $x\in[N]$ independently red/blue with probability $1/2$. For each $k$--term arithmetic progression $P=\{a,a+d,\dots,a+(k-1)d\}\subseteq[N]$, let $E_P$ be the "bad" event that $P$ is monochromatic. Then \[ \mathbb{P}(E_P)=2\cdot (1/2)^k = 2^{1-k}=:p. \]

We use the symmetric Lovász Local Lemma. Two events $E_P,E_Q$ are independent unless the progressions $P$ and $Q$ share at least one element. So we need an upper bound on how many $k$--APs in $[N]$ can intersect a fixed $k$--AP.

Claim. Each integer $x\in[N]$ lies in at most \[ T := k\,\lfloor\frac{N-1}{k-1}\rfloor \] many $k$--term arithmetic progressions in $[N]$.

Proof of claim. Any $k$--AP in $[N]$ has common difference $d$ satisfying $1\le d\le \lfloor (N-1)/(k-1)\rfloor$ (since $a+(k-1)d\le N$). Fix such a $d$. If $x$ is the $j$th term of the progression ($j\in\{0,1,\dots,k-1\}$), then the start is forced to be $a=x-jd$. Thus for each $d$ there are at most $k$ progressions containing $x$ (one for each possible $j$), and there are at most $\lfloor (N-1)/(k-1)\rfloor$ possible values of $d$. Multiplying gives the claim. $\triangleleft$

Now fix a progression $P$. It has $k$ elements, and for each element $x\in P$ there are at most $T$ progressions containing $x$. Therefore the number of progressions $Q$ with $Q\cap P\ne\emptyset$ is at most $kT$ (this overcounts but is an upper bound). Hence each event $E_P$ depends on at most \[ D := kT-1 \] other events. In particular, \[ D+1 \le kT = k^2\lfloor\frac{N-1}{k-1}\rfloor \le \frac{k^2 N}{k-1}. \]

The symmetric Lovász Local Lemma states that if \[ e\,p\,(D+1) \le 1, \] then $\mathbb{P}(\bigcap_P \overline{E_P})>0$, i.e. a coloring exists with no bad events. A sufficient condition is \[ e\cdot 2^{1-k}\cdot \frac{k^2 N}{k-1} \le 1, \] which is exactly $N\le \frac{(k-1)2^{k-1}}{e k^2}$. Thus such a coloring exists, implying $W(k)>N$. ∎

Remark: Proposition Proposition (Local Lemma bound) only yields $\log W(k)=\Omega(k)$, hence $W(k)^{1/k}$ stays bounded (indeed the lower bound gives $W(k)^{1/k}\gtrsim 2$). This illustrates the size of the gap to the desired $W(k)^{1/k}\to\infty$, which would require $\log W(k)/k\to\infty$.

5) VERIFICATION / EDGE CASES

  • The dependency bound in Proposition Proposition (Local Lemma bound) is intentionally crude but valid: counting progressions through each point and summing over the $k$ points of $P$ yields an upper bound on intersecting progressions.

  • For $k=1,2$ the definition of $W(k)$ is trivial; Proposition Proposition (Local Lemma bound) assumes $k\ge 3$ so that the "$k$--AP" events are genuinely constrained.

  • The tiny-case verification $W(3)=9$ was checked both by an explicit $[8]$ coloring and by exhaustive enumeration on $[9]$.

6) FINAL

UNRESOLVED

  1. Farthest point reached. I proved the standard Lovász Local Lemma lower bound $W(k)\ge c\,2^k/k$ (Proposition Proposition (Local Lemma bound)) and sanity-checked $W(3)=9$.

  2. Best partial lemma. Proposition Proposition (Local Lemma bound): existence of $2$--colorings of length $\asymp 2^k/k$ with no monochromatic $k$--AP.

  3. Smallest missing step. To prove $W(k)^{1/k}\to\infty$ one needs a qualitatively new lower bound giving $\log W(k)\gg k\log k$ (or stronger), far beyond Local Lemma/exponential constructions.

  4. Completion estimate. The current best general lower bounds are exponential in $k$; obtaining super--exponential growth appears to require new structural/algebraic constructions or breakthroughs in quantitative Ramsey/additive combinatorics.

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