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Statement

Among the "Unproved Conjectures" of the appendix (pp. 158--159):

• Conjecture 1 (p. 158): "It is possible to replace the constant $3\cdot6^{1/2}$ in Theorem I by the constant 2."
• Conjecture 3 (p. 158): "$F(0)>0$ for $k>2p^{1/2}$, where $p$ is not necessarily a prime." The text before it says "For composite moduli Theorem I and II cease to be true. It is however reasonable to formulate" the conjecture, and after it (p. 159): "This conjecture may also be true for finite abelian groups of composite order $p$, and possibly even, mutatis mutandis, for non-abelian groups."

Here $F(0)$ counts solutions of $e_1a_1+\cdots+e_ka_k\equiv0\pmod p$ with $e_i\in\{0,1\}$ for $k$ distinct nonzero residues; the all-zero choice is a solution, so the conjecture is read, as the later literature reads it, as asking for a nonempty zero-sum subset. Conjecture 2 (p. 158) compares $G(S_k)$ with $G(S_k^*)$ for the alternating sequence and would imply Conjecture 1; Conjecture 4 (p. 159) concerns choosing one residue from each of $s$ blocks and, the paper notes as it goes to press, follows from a result of Scherk.

Source. P. Erdős and H. Heilbronn, On the addition of residue classes mod $p$, Acta Arith. 9 (1964), no. 2, 149--159, DOI 10.4064/aa-9-2-149-159; the appendix on printed pp. 158--159 (PDF pp. 10--11 of the eleven-page scan), read on the page images.

Read depth. Claims checked: Conjectures 1--4 and the two sentences around Conjecture 3 were read clause by clause on the page images.

Proof pointer

None; conjectures. Conjecture 3 with an unspecified constant is Szemerédi's 1970 theorem for all finite abelian groups; with the constant $2$ it holds for prime $p$ by Olson (1968) and, in the sharper form $\sqrt{2p}$, by Balandraud's Theorem 9; for arbitrary finite abelian groups Hamidoune and Zémor prove $\sqrt{2n}+O(n^{1/3}\ln n)$. Olson's paper (J. Combinatorial Theory 5 (1968), 45--52), of which no file is held, is filed as olson_1968_addition_theorem_modulo; its Theorem 1, "If $s>(4p-3)^{1/2}$, then $r=p$", with the abstract's "we verify a conjecture of P. Erdös and H. Heilbronn: every residue class is represented if $s>2p^{1/2}$", is on printed p. 45 (PDF p. 1), located here on the text layer of that page on 2026-09-22 and paged on theorem_1.

Dependencies

None.

Bears on

• Problem 540: the origin; the site's "A conjecture of Erdős and Heilbronn [ErHe64]". The problem's "$\gg N^{1/2}$" is the weakening with an unspecified constant, and the paper's own constant is $2$ (the $\sqrt2$ of the site's "Erdős speculated" appears in Erdős's 1965 and 1973 problem papers, not here).