A resolution of Erd\H{o}s Problem #190 via Erd\H{o}s-Lov\'asz, 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. Erd\H{o}s and Graham asked whether $H(k)^{1/k}/k \to \infty$ (Problem #190 of the Erd\H{o}s 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 $H(k)^{1/k}/k = \Omega(k/\log k)$ and $\lim_{k\to\infty} H(k)^{1/k}/k = \infty$, resolving the positive direction of the Erd\H{o}s-Graham question. The argument combines three standard ingredients -- the symmetric Lov\'asz 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 Erd\H{o}s-Lov\'asz had fixed $r$; the improvement here is that the $r^{k-1}$ base dominates once one allows the color count $r_0 = \lfloor k / \log k \rfloor$ to grow with $k$. No matching upper bound on $H(k)^{1/k}/k$ is known.