Problem 1211

Let $\mathbb{N}=A\cup B$ where $A$ and $B$ are disjoint. Let $S(A)$ be those integers which are the sum of finitely many distinct elements of $A$ , and similarly for $S(B)$ .

\overline{\delta}(X)=\limsup_{x\to \infty}\frac{1}{\log x}\sum_{\substack{n\in X\\ n<x}}\frac{1}{n}

is the upper logarithmic density of $X$ then how large must

\max(\overline{\delta}(S(A)),\overline{\delta}(S(B)))

be?

Source: erdosproblems.com/1211

Number theory
Ramsey theory

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An accepted solution exists. Settled in another form, for example when its parts resolve differently or the question is open-ended.

Solved, in the site's label for a determination question. The status-defining source is Theorem 1 of Conlon, Fox and Pham: for every partition of $\mathbb N$ into two classes the larger upper logarithmic density of the subset sums is at least $(2+\sqrt3)/4\approx0.93301$ , and the coloring by the parity of $\lfloor\log_{2+\sqrt3}\log n\rfloor$ attains it, so the minimum is $c_2=(2+\sqrt3)/4$ . The paper appeared in Mathematika 68 (2022), no. 4, 1292--1301 (published online 10 October 2022, per the Crossref record), a refereed journal; the text cited here is the arXiv version v3 of 22 September 2022, and the journal text was not compared. The site accepted the result (SOLVED, last edited 8 April 2026), its curator crediting the paper with the value in the commentary. The claim page Conlon, Fox and Pham 2021 records the theorem, its postings and the acceptance evidence; the frontmatter standing is derived from it. Erdős's expectation $c>1/2$ holds, and his claim that the value cannot exceed $3/4$ was wrong, his example giving $14/15$ .

CFP22: Conlon, D., Fox, J. and Pham, H. T., The upper logarithmic density of monochromatic subset sums. Mathematika 68 (2022), no. 4, 1292--1301; DOI 10.1112/mtk.12167; arXiv:2105.15195v3 (22 September 2022, 9 pp.). Theorem 1 on p. 2, the definitions on pp. 1--2, Lemma 2 and Theorem 3 on p. 3, Remark 5 on p. 4, Conjecture 10 on p. 9 (preprint pages). Library home: Conlon, Fox and Pham 2022; result page Conlon, Fox and Pham 2022, Theorem 1.
Er80: Erdős, P., A survey of problems in combinatorial number theory. Ann. Discrete Math. 6 (1980), 89--115; printed p. 113. Library home: Erdos 1980.
Er82c: Erdős, P., Miscellaneous problems in number theory. Proceedings of the Eleventh Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, 1981), Congr. Numer. 34 (1982), 25--45; Part II, Theorem 3 and display (23) on printed p. 38 and displays (24)--(26) on p. 39. Library home: Erdős 1982.
Er82d: Erdős, P., Some new problems and results in number theory. In Number theory (Mysore, 1981), Lecture Notes in Math. 938, Springer (1982), 50--74; the site's reference text. The Rényi archive's scan (users.renyi.hu/~p_erdos/1982-32.pdf, 25 pp.) carries the passage at printed p. 54 (PDF p. 5). Library home: Erdős 1982 (the card carries the row for this problem). It is the paper's reference [3].
CFP21: Conlon, D., Fox, J. and Pham, H. T., Subset sums, completeness and colorings. arXiv:2104.14766v1 (2021); its Theorem 6.1 is the input Theorem 3 of [CFP22]. Library home: Conlon 2021.
SzVu06: Szemerédi, E. and Vu, V. H., Long arithmetic progressions in sumsets: thresholds and bounds. J. Amer. Math. Soc. 19 (2006), 119--169; its Theorem 7.1 is the published alternative input named in Remark 5 of [CFP22]. Not held.

Problem 1211

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