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The sumset of the subsets $A$ and $B$ of an additively written group is defined by $A+B:=\{a+b\colon a\in A,\ b\in B\}$. The basic idea to add sets has been around since Cauchy at least.

Erdős and Heilbronn considered a variation where equal summands are excluded from consideration. This version of set addition was further generalized to consider restricted sumsets where the sums allowed are taken "along the edges of a graph": that is, given a set $E\subseteq A\times B$, we let $A\stackrel{E}{+}B:=\{a+b\colon a\in A,\ b\in B,\ (a,b)\in E\}$. Perhaps, the most famous result where restricted sumsets play a central role is the Balog-Szemerédi-Gowers theorem.

I am interested in the historical component of the story.

  • When and where restricted sumsets of a general form (that is, the addition of sets along a graph) were introduced?
  • Who has coined the name "restricted sumset"?
  • Where does the notation $A\stackrel{E}{+}B$ originate from?
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    $\begingroup$ In this preprint, arxiv.org/pdf/0905.0135.pdf, they seem to say that the answer to your first question is the original sum-product paper of Erdős–Szemerédi ("On sums and products of integers"). I believe equation (3) in the Erdős–Szemerédi paper is what they mean. Erdős–Szemerédi do not use either the terminology restricted sumset, or the sumset notation, however. $\endgroup$
    – Anurag Sahay
    Feb 20, 2022 at 21:10
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    $\begingroup$ Here's a link to the Erdős–Szemerédi paper: citeseerx.ist.psu.edu/viewdoc/… $\endgroup$
    – Anurag Sahay
    Feb 20, 2022 at 21:11
  • $\begingroup$ I suppose a related question (whose answer I would guess you know already) is who introduced the word sumset and the notation $A+B$ for sumsets. That would give a lower bound for the year for your other two questions that might make it easier to search for the answer. $\endgroup$
    – Anurag Sahay
    Feb 20, 2022 at 22:16

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