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Claim. There is a constant $c<1$ such that $A(N)<cN$ for all large $N$: every $A\subseteq\{1,\ldots,N\}$ with $|A|\ge(1-\delta)N$, for a fixed small $\delta>0$ and $N$ large, has a subset with reciprocal sum one. This refutes the expectation of Erdős and Graham, printed in their 1980 monograph, that $A(N)=N+o(N)$.

Covers. The qualitative bound only: the existence of some $c<1$, with no value of $c$ and no asymptotic. The value disproved records what the bound decides: the conjectured estimate $A(N)=N+o(N)$ of the monograph, which the 1999 booklet's form of the problem asks about directly and the site's question leaves to its commentary, is false. The asymptotic $A(N)=(1-1/e+o(1))N$ is the full claim Liu and Sawhney's Theorem 1.3, which implies this bound.

Acceptance. Croot's paper, On a coloring conjecture about unit fractions, Annals of Mathematics (2) 157 (2003), no. 2, 545–556, is refereed, and its publication date, 1 March 2003 in the DOI record of the March 2003 issue, dates this page; the arXiv posting of 24 November 2003 is the published version uploaded later. The refereed paper does not state the bound, so the page lists no refereed evidence: its Main Theorem is a unit-subsum criterion for a set of smooth integers in a short range whose reciprocal mass exceeds six. The site's curator, Thomas Bloom, credits Croot's work with the disproof in the problem's commentary, independently of the author, and Liu and Sawhney write in their refereed paper that the bound follows from Croot's work, so the only refereed statement of the bound is theirs, not Croot's; neither writes the deduction out, and the corpus has not compiled it. The page records that attribution as made, not a compiled result.