Erdős Problem #348

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 source explicitly allows a nondecreasing sequence, hence repeated values. Deleting terms and taking distinct summands are interpreted by occurrences (indices), so two equal values at different positions remain two available terms. This convention differs from treating the sequence as a set of distinct values. [S1] [S2]

Statement recorded in the supplied source.

For what values of $0\leq m<n$ is there a complete sequence $A=\{a_1\leq a_2\leq \cdots\}$ of integers such that

• $A$ remains complete after removing any $m$ elements, but
• $A$ is not complete after removing any $n$ elements?

[S1]

Scope and interpretation.

The occurrence convention follows the source’s nondecreasing-sequence notation. A version using only distinct values would be a separate formulation. [S1]

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

Short English statement

Which pairs of deletion counts can occur for a sequence that survives every deletion of the smaller size but none of the larger size?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2066 (EP-348), 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 348, https://www.erdosproblems.com/348. Reference from the supplied record; the live problem page was not independently checked for this compilation.