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Claim. For every partition $\mathbb N=A\sqcup B$,

$$\max\bigl(\overline{\delta}(S(A)),\overline{\delta}(S(B))\bigr)\ge\frac{2+\sqrt3}4\approx0.93301,$$

where $S(X)$ is the set of sums of finitely many distinct elements of $X$ and $\overline{\delta}$ the upper logarithmic density, and the partition into the $n$ with $\lfloor\log_b\log n\rfloor$ even and the $n$ with it odd, $b=2+\sqrt3$, has maximum equal to $(2+\sqrt3)/4$. So the least possible value of the maximum, the quantity Problem 1211 asks for, is $c_2=(2+\sqrt3)/4$. This is the case $r=2$ of Theorem 1 of the library's source card, which bounds the analogous constant $c_r$ for every number $r\ge2$ of classes and proves the bound tight for $r=2$. The upper bound generalizes Erdős's block coloring; the lower bound rests on the paper's Lemma 2, a statement about subset sums of a class of a partition of $[N,eN)$ filling an interval $[CN,C'N^2]$, turned into the density bound through an auxiliary coloring and the Brouwer fixed-point theorem. The site's question asks how large the maximum must be, and the theorem answers it with the exact value, so the claim settles the whole question; the tightness of the bound for $r\ge3$ is the paper's Conjecture 10 and is not part of this problem.

Depends on. Nothing in this wiki; the result is the paper's own theorem.

Acceptance. Reviewed: the site's curator, T. F. Bloom, labels the problem SOLVED and credits Conlon, Fox and Pham in the problem's commentary with the value $c=(2+\sqrt3)/4$ and with the coloring that attains it (page last edited 8 April 2026); the curator is independent of the authors, and the community database records the problem solved (last updated 4 April 2026). Refereed: Mathematika 68 (2022), no. 4, 1292--1301, published online 10 October 2022 (per the Crossref record). The text cited is arXiv:2105.15195v3 (22 September 2022); the journal text was not compared, so every locator is a preprint page. The search (arXiv, Crossref, OpenAlex) found no dispute of the theorem and no second determination of $c_2$.

Read depth. Theorem 1, the definitions and the upper-bound argument on pp. 1--2 of the preprint were checked, as were the statements of Lemma 2, Theorem 3, Remark 5 and Conjecture 10; the lower-bound proof (Section 3, pp. 4--8) was not checked, and nothing is independently reviewed in this corpus. The problem page checks by an elementary block count that Erdős's own example has both densities equal to $14/15$, as the paper states. The page is dated by the first arXiv posting, v1 of 31 May 2021.