A resolution of Erdős Problem #190 via Erdős-Lovász, BCT, and Baker-Harman-Pintz

Abstract: Let H(k) be the smallest N such that every finite coloring of [N] contains a monochromatic or rainbow k-term arithmetic progression. Erdős and Graham asked whether $H(k)^{1/k}/k \to \infty$ (Problem #190 of the Erdős Problems database). We prove that there is an absolute constant $k_0 \ge 2$ such that for all $k \ge k_0$, [ H(k)^{1/k}/k \ge (1/e - \varepsilon(k)) \cdot k/\log k, \qquad \varepsilon(k) = O(k^{-0.475} \log k) \to 0 \text{ as } k \to \infty; ] in particular $H(k)^{1/k}/k = Ω(k/\log k)$ and $\lim_{k\to\infty} H(k)^{1/k}/k = \infty$, resolving the positive direction of the Erdős-Graham question. The argument combines three standard ingredients -- the symmetric Lovász Local Lemma applied to the k-AP hypergraph on $[N]$, the restricted form of the Blankenship-Cummings-Taranchuk recurrence, and the Baker-Harman-Pintz prime-gap theorem -- together with the pigeonhole reduction $H(k) \ge W(k-1,k)$, and uses BHP as the only analytic black box. Previous applications of Erdős-Lovász had fixed $r$; the improvement here is that the $r<sup>{k-1}$ base dominates once one allows the color count $r_0 = \lfloor k / \log k \rfloor$ to grow with $k$. No matching upper bound on $H(k)<sup>{1/k}/k$ is known.

• The paper resolves the positive direction of Erdős–Graham Problem #190 by proving that H(k)^{1/k}/k ≥ (1/e−ε(k))k/log k, so the ratio diverges.
• The proof combines a uniform Erdős–Lovász bound at r₀=⌊k/log k⌋, the BCT blow-up recurrence, and a Baker–Harman–Pintz prime near k to obtain H(k)^{1/k}/k=Ω(k/log k).
• The analysis shows that 1/e is optimal within this proof strategy, while no matching upper bound or definitive asymptotic rate for H(k) is currently known.

The paper resolves the positive direction of a question of Erdős and Graham on the canonical Ramsey function $H(k)$, the least $N$ such that every finite coloring of $[N]$ contains either a monochromatic or a rainbow $k$-term arithmetic progression (k-AP). The main result is that for all sufficiently large $k$,

$$\frac{H(k)^{1/k}}{k}\;\ge\;\Bigl(\frac{1}{e}-\varepsilon(k)\Bigr)\frac{k}{\log k},\qquad \varepsilon(k)=O(k^{-0.475}\log k),$$

and hence $H(k)^{1/k}/k\to\infty$, answering Problem #190 in the Erdős Problems database (2604.20588). No matching upper bound on $H(k)^{1/k}/k$ is known, and the paper is explicit on this point: the qualitative divergence is established, but the true asymptotic rate remains undetermined.

Background and the main theorem

The Erdős–Graham question asks whether $H(k)^{1/k}/k\to\infty$. Prior lower bounds on $H(k)$ were obtained by composing lower bounds on van der Waerden numbers $N$0 with the pigeonhole reduction $N$1: any $N$2-coloring avoiding monochromatic k-APs automatically avoids rainbow k-APs, since it uses too few colors. The available bases were Berlekamp's $N$3 (for $N$4 prime), the Erdős–Lovász Local Lemma bound $N$5, and Hunter's 2025 exponential improvement for fixed $N$6. Composing any of these at a fixed color count $N$7 yields only a bounded ratio: the Berlekamp route gives $N$8, while a direct application of Erdős–Lovász at $N$9 gives a ratio tending to $[N]$0 from below. Hunter's route gives at most $[N]$1, and even that only conditionally on a uniform extension not proved in Hunter's paper.

The key observation is that the Erdős–Lovász base $[N]$2 becomes super-exponential once the color count is allowed to grow with $[N]$3. The paper takes $[N]$4 and then iterates the Blankenship–Cummings–Taranchuk (BCT) blow-up recurrence to climb from $[N]$5 up to $[N]$6, a prime in $[N]$7 supplied by the Baker–Harman–Pintz (BHP) prime-gap theorem. The final rate is polynomial in $[N]$8.

The three ingredients

The proof rests on three standard results, each of which the paper re-derives or states precisely.

Erdős–Lovász base. Via the symmetric Lovász Local Lemma (whose proof is reproduced in full, including the $[N]$9 edge case), applied to the k-AP hypergraph on $k$0 with a uniformly random $k$1-coloring, the paper obtains $k$2 for all $k$3, $k$4. The constant 16 is acknowledged as non-optimal (Alon–Spencer give $k$5), and the threshold $k$6 is likewise not sharp (any $k$7 works). What matters is uniformity in $k$8: the constant and threshold are absolute, so the bound applies at the growing $k$9, which the paper identifies as essential.

BCT recurrence. The restricted form used is: for prime $k$0, $k$1, and $k$2, one has $k$3. The paper gives a self-contained proof by a block blow-up construction: each of the $k$4 positions in a block inherits the base color except one reserved position, which receives the new color $k$5. The hypothesis $k$6 is used only in the case $k$7, where within-block offsets cycle through all residues mod $k$8 over $k$9 consecutive AP terms; the hypothesis $$\frac{H(k)^{1/k}}{k}\;\ge\;\Bigl(\frac{1}{e}-\varepsilon(k)\Bigr)\frac{k}{\log k},\qquad \varepsilon(k)=O(k^{-0.475}\log k),$$0 ensures the reserved offset lies within the block. Both hypotheses hold in the application.

BHP prime gap. For $$\frac{H(k)^{1/k}}{k}\;\ge\;\Bigl(\frac{1}{e}-\varepsilon(k)\Bigr)\frac{k}{\log k},\qquad \varepsilon(k)=O(k^{-0.475}\log k),$$1, the interval $$\frac{H(k)^{1/k}}{k}\;\ge\;\Bigl(\frac{1}{e}-\varepsilon(k)\Bigr)\frac{k}{\log k},\qquad \varepsilon(k)=O(k^{-0.475}\log k),$$2 contains a prime. Applied at $$\frac{H(k)^{1/k}}{k}\;\ge\;\Bigl(\frac{1}{e}-\varepsilon(k)\Bigr)\frac{k}{\log k},\qquad \varepsilon(k)=O(k^{-0.475}\log k),$$3, this yields a prime $$\frac{H(k)^{1/k}}{k}\;\ge\;\Bigl(\frac{1}{e}-\varepsilon(k)\Bigr)\frac{k}{\log k},\qquad \varepsilon(k)=O(k^{-0.475}\log k),$$4 with $$\frac{H(k)^{1/k}}{k}\;\ge\;\Bigl(\frac{1}{e}-\varepsilon(k)\Bigr)\frac{k}{\log k},\qquad \varepsilon(k)=O(k^{-0.475}\log k),$$5. BHP is the only analytic black box in the argument.

The proof and the rate

The chain is: (i) $$\frac{H(k)^{1/k}}{k}\;\ge\;\Bigl(\frac{1}{e}-\varepsilon(k)\Bigr)\frac{k}{\log k},\qquad \varepsilon(k)=O(k^{-0.475}\log k),$$6 by pigeonhole; (ii) $$\frac{H(k)^{1/k}}{k}\;\ge\;\Bigl(\frac{1}{e}-\varepsilon(k)\Bigr)\frac{k}{\log k},\qquad \varepsilon(k)=O(k^{-0.475}\log k),$$7 by Erdős–Lovász; (iii) iterating the BCT recurrence at the fixed prime $$\frac{H(k)^{1/k}}{k}\;\ge\;\Bigl(\frac{1}{e}-\varepsilon(k)\Bigr)\frac{k}{\log k},\qquad \varepsilon(k)=O(k^{-0.475}\log k),$$8 over $$\frac{H(k)^{1/k}}{k}\;\ge\;\Bigl(\frac{1}{e}-\varepsilon(k)\Bigr)\frac{k}{\log k},\qquad \varepsilon(k)=O(k^{-0.475}\log k),$$9 gives $H(k)^{1/k}/k\to\infty$0; (iv) monotonicity of $H(k)^{1/k}/k\to\infty$1 in $H(k)^{1/k}/k\to\infty$2 gives $H(k)^{1/k}/k\to\infty$3. Hence

$H(k)^{1/k}/k\to\infty$4

Taking $H(k)^{1/k}/k\to\infty$5-th roots and dividing by $H(k)^{1/k}/k\to\infty$6, three factors are evaluated one-sidedly. The $H(k)^{1/k}/k\to\infty$7 factor is $H(k)^{1/k}/k\to\infty$8; the factor $H(k)^{1/k}/k\to\infty$9 equals $H(k)^{1/k}/k$0; and the factor $H(k)^{1/k}/k$1 is bounded below via $H(k)^{1/k}/k$2 and $H(k)^{1/k}/k$3, giving $H(k)^{1/k}/k$4. The separation $H(k)^{1/k}/k$5, needed to ensure $H(k)^{1/k}/k$6, is verified by showing $H(k)^{1/k}/k$7 is positive and increasing for $H(k)^{1/k}/k$8 (with $H(k)^{1/k}/k$9). The resulting error rate $H(k)^{1/k}/k\to\infty$0 is dominated by the BHP prime-gap contribution; the $H(k)^{1/k}/k\to\infty$1 loss from the BCT iteration length is exact, not an error. The paper notes that all constants are absolute and effective in principle, though it does not optimize them, and an effective form of BHP would be needed to make $H(k)^{1/k}/k\to\infty$2 explicit.

Optimality within the method

The paper analyzes what the method yields for a general base $H(k)^{1/k}/k\to\infty$3, obtaining $H(k)^{1/k}/k\to\infty$4. Optimizing gives $H(k)^{1/k}/k\to\infty$5 and $H(k)^{1/k}/k\to\infty$6, so the constant $H(k)^{1/k}/k\to\infty$7 is sharp within this EL + BCT + BHP chain: no choice of $H(k)^{1/k}/k\to\infty$8 can improve on $H(k)^{1/k}/k\to\infty$9 by this route. The analysis is instructive in both directions: any fixed multiple $H(k)$0 gives the same order with constant $H(k)$1; choices $H(k)$2 give divergent but smaller rates (e.g., $H(k)$3 gives $H(k)$4); and pushing $H(k)$5 to $H(k)$6 collapses the bound to $H(k)$7. Improving on $H(k)$8 would require a sharper Erdős–Lovász base, a prime-gap exponent below $H(k)$9, or a replacement for the BCT step.

Limitations and open questions

The paper is candid about what it does not settle. First, no asymptotic upper bound on $N$00, or on $N$01 from above, is known; the divergence result leaves open whether the true rate is $N$02, polynomially larger, or in between. Second, the constant $N$03 is method-dependent, and its global tightness is an open question. Third, the conditional Hunter-route bound $N$04 with $N$05 is strictly weaker than the present rate and rests on a uniform extension of Hunter's fixed-$N$06 result that is not proved there. Finally, the threshold $N$07 depends on the (non-explicit) BHP threshold, so the theorem is asymptotic rather than effective.

Conclusion

The paper establishes $N$08 and thereby resolves the Erdős–Graham question affirmatively, by combining the Erdős–Lovász Local Lemma at a growing color count $N$09, the BCT recurrence iterated up to a BHP prime $N$10, and the pigeonhole reduction $N$11. The contribution is the identification of the growing-$N$12 regime, which converts previously bounded ratios into a polynomial rate, together with the demonstration that $N$13 is the optimal constant achievable within this particular chain of inequalities. The principal open problem is an upper bound on $N$14 commensurate with the new lower bound.