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Claim. Section 3 of Vsevolod F. Lev, Reconstructing integer sets from their representation functions, Electron. J. Combin. 11 (2004), no. 1, #R78, builds a single perfect difference set greedily. Step $n$ adds $z_n$ and $z_n+d_n$, where $d_n$ is the least difference not yet represented and $z_n$ is chosen so that neither number is already in the set and no nontrivial equation $a_1-a_2=a_3-a_4$ arises among its elements. The paper observes that these conditions exclude $O(n^3)$ values of $z_n$ and that $d_n=O(n^2)$, so the numbers added at step $n$ are $O(n^3)$, which it states as the $n$th element of the set being $O(n^3)$. For the $a_m$ of Problem 1194 this gives $a_m\ll m^3$: the $d_n$ strictly increase, so a difference $m$ first represented at step $s$ satisfies $d_s\le m$ and hence $s\le m+1$, and both elements representing $m$ were added by step $s$, so $a_m=O(s^3)=O(m^3)$. The site's remarks record this bound for the greedy construction and credit its details to Lev. The source card is lev_2004_reconstructing_integer_sets_representation_functions.

Covers. An upper bound: some perfect difference set has $a_n\ll n^3$, so $a_n/n$ need not grow faster than $n^2$. It does not determine how fast $a_n/n$ must grow.

Depends on. No page of this wiki.

Acceptance. Refereed: the construction and its bound are in Section 3 of the paper in the Electronic Journal of Combinatorics, published 2004-11-03; the step from the paper's bound to $a_m$ is spelled out above. The site's remarks credit the construction, but the site labels the problem OPEN, so the remarks are not acceptance.