Erdős Problem #117

Definitions and mathematical statement

Shared notation.

Unless a section says otherwise, ${\mathbb{N}}=\{1,2,\ldots\}$, ${\mathbb{Z}}$ is the ring of integers, ${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$ are the rational, real, and complex fields, $[N]=\{1,\ldots,\lfloor N\rfloor\}$, and $\log$ is the natural logarithm. For real functions of a parameter tending to infinity, $f\ll g$ and $f=O(g)$ mean $|f|\leq Cg$ eventually for some fixed $C$; $f\gg g$ reverses this comparison; $f=o(g)$ means $f/g\to0$; $f\sim g$ means $f/g\to1$; and $f\asymp g$ means both order bounds. A subscript on $O$ or $\ll$ specifies allowed constant dependence. The expression $n^{o(1)}$ in an upper bound means growth smaller than $n^\epsilon$ for each fixed $\epsilon>0$. Local definitions govern any exception, finite-field convention, or use of the same symbol for another object. An asymptotic claim is not automatically an effective algorithm.

An abelian subgroup has every pair of its elements commuting. A cover is a union of subgroups equal to the whole group, not a partition. The hypothesis bounds the size of a pairwise noncommuting subset; the group itself is not assumed finite. [S1] [S2]

Statement recorded in the supplied source.

Let $h(n)$ be minimal such that any group $G$ with the property that any subset of $>n$ elements contains some $x\neq y$ such that $xy=yx$ can be covered by at most $h(n)$ many Abelian subgroups. Estimate $h(n)$ as well as possible. [S1]

Transcription note. Only unambiguous escape/formatting damage and nonmathematical serialization debris were repaired in the displayed source statement.

Short English statement

If sufficiently large subsets of a group always contain two commuting elements, how many abelian subgroups suffice to cover the entire group?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 1938 (EP-117), statement and background. The embedded literature-review date is 2026-08-17; that is the archive’s review date, not a fresh certification. Dataset landing page: https://huggingface.co/datasets/ulamai/UnsolvedMath.

• S2 T. F. Bloom, Erdős Problems, Problem 117, https://www.erdosproblems.com/117. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 László Pyber, The Number of Pairwise Non-Commuting Elements and the Index of the Centre in a Finite Group, Journal of the London Mathematical Society s2-35 (1987), 287-295,

  \nolinkurl{doi:10.1112/jlms/s2-35.2.287.}

  https://doi.org/10.1112/jlms/s2-35.2.287. Bibliographic information from the supplied record.

• S4 Erdős, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10. Bibliographic information from the supplied record.