Erdős Problem #665

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.

The sets $A_i$ are blocks. Saying that each unordered pair belongs to exactly one block makes the pair sets $\binom{A_i}{2}$ a partition of $\binom{[n]}2$. The desired bound uses one absolute constant $C$ for all sufficiently large ground-set sizes; the number and sizes of blocks may vary with $n$. [S1] [S2]

Statement recorded in the supplied source.

A pairwise balanced design for $\{1,\ldots,n\}$ is a collection of sets $A_1,\ldots,A_m\subseteq \{1,\ldots,n\}$ such that $2\leq \lvert A_i\rvert <n$ and every pair of distinct elements $x,y\in \{1,\ldots,n\}$ is contained in exactly one $A_i$. Is there a constant $C>0$ and, for all large $n$, a pairwise balanced design such that $ \lvert A_i\rvert > n^{1/2}-C $ for all $1\leq i\leq m$? [S1]

Residual task reported by the archive.

The embedded review (2026-08-17) records: Determine h(n) or connect it unconditionally to prime gaps. [S1]

Short English statement

For every sufficiently large ground-set size, can each pair be covered exactly once by proper blocks all within a fixed additive amount below square-root size?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2241 (EP-665), 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 665, https://www.erdosproblems.com/665. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 Erdős, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10. Bibliographic information from the supplied record.

• S4 Erdős, P. and Larson, J., On pairwise balanced block designs with the sizes of blocks as uniform as possible. Annals of Discrete Mathematics (1982), 129-134. Bibliographic information from the supplied record.

• S5 S. S. Shrikhande and N. M. Singhi, On a problem of Erdős and Larson. Combinatorica (1985), 351-358. Bibliographic information from the supplied record.