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The claim. For every prime $p$, a zero-sum free subset of $\mathbb Z/p\mathbb Z$ of the largest possible size has exactly $k$ elements, where $k$ is the greatest integer with $k(k+1)/2<p$. This is Theorem 9 of É. Balandraud, An addition theorem and maximal zero-sum free sets in $\mathbb Z/p\mathbb Z$, arXiv:0907.3492v1 (20 July 2009), p. 16, published in Israel J. Math. 188 (2012), no. 1, 405--429, with an erratum, ibid. 192 (2012), no. 2, 1009--1010; the labels are those of the arXiv version. Paged as Theorem 9 of Balandraud (2012). Every subset of $\mathbb Z/p\mathbb Z$ with more than $k$ elements therefore has a nonempty zero-sum subset, so for prime $N$ the threshold of Problem 540 is $\sqrt{2p}+O(1)$, the constant $\sqrt2$ that Erdős suggested and Selfridge's 1976 conjecture. The theorem is deduced from the paper's addition theorem for subsums (Theorem 5).

Covers. Prime $N$, with the exact threshold of Theorem 9. Composite $N$ is not covered.

Acceptance. Refereed: the journal publication. Reviewed: the site's curator, Thomas Bloom, who is independent of the author, labels the problem PROVED (LEAN) and his commentary credits this paper with Selfridge's conjecture for prime $N$. The erratum has not been compared with the arXiv version.

Depends on. No page of this wiki.