Snevily's conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $G$ be a finite abelian group, written additively. For equal-sized sets $A,B\subseteq G$, the addition submatrix has rows $a\in A$, columns $b\in B$, and entry $a+b$. A Latin transversal selects one cell in every row and column with all selected entries distinct. For a subgroup $H\le G$, $x+H$ denotes its coset.

Statement recorded in the supplied source.

For every odd-order $G$ and sets $|A|=|B|=k$, do there exist enumerations $A=\{a_1,\ldots,a_k\}$, $B=\{b_1,\ldots,b_k\}$ with the $k$ sums $a_i+b_i$ pairwise different? This is equivalently the Latin-transversal assertion for every square addition submatrix.

The source additionally poses the even-order extension: for $G=\mathbb Z/(2n)\mathbb Z$, must every such submatrix have a Latin transversal unless its row and column sets are cosets $A=x+H$, $B=y+H$ of one even-order subgroup $H$? It also asks whether this obstruction-only assertion extends to every finite abelian group, with $H$ an even-order cyclic subgroup. [S2]

Short English statement

Pair two equal-sized subsets of an odd-order abelian group with distinct sums; also consider the source’s even-order extension excluding translated even cyclic subgroup tables. The odd-order assertion is proved by Arsovski, as recorded in the cited exposition. [S3]

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3385 (OPG-655). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Snevily's conjecture, OPG-655. Original problem statement, full mathematical discussion, bibliography and comments. https://www.openproblemgarden.org/op/snevilys_conjecture.
• S3 Gergely Harcos, Gyula Károlyi and Géza Kós, Remarks to Arsovski’s proof of Snevily’s conjecture, arXiv:1004.0253v2 (2010). Introduction and proof of Snevily’s conjecture. https://arxiv.org/abs/1004.0253v2.