Erdős Problem #734

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.

Every unordered pair of ground-set elements belongs to exactly one block. Nontrivial means the blocks have sizes at least two and strictly below the whole ground-set size. The constant controlling the number of blocks of any one size must be independent of both the size and $n$. [S1] [S2]

Statement recorded in the supplied source.

Find, for all large $n$, a non-trivial pairwise balanced block design $A_1,\ldots,A_m\subseteq \{1,\ldots,n\}$ such that, for all $t$, there are $O(n^{1/2})$ many $i$ such that $\lvert A_i\rvert=t$. [S1]

Residual task reported by the archive.

The embedded review (2026-08-17) records: Construct PBDs with every size used O(sqrt(n)) times. [S1]

Short English statement

Can every sufficiently large ground set admit a nontrivial pairwise balanced design with only square-root many blocks of any one size?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2285 (EP-734), 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 734, https://www.erdosproblems.com/734. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. Bibliographic information from the supplied record.

• S4 de Bruijn, N. G. and Erdős, P., On a combinatorial problem. Nederl. Akad. Wetensch., Proc. (1948), 1277--1279 = Indagationes Math. 10, 421--423. Bibliographic information from the supplied record.