Erdős Problem #616

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 $r$-uniform hypergraph is a family of distinct $r$-element subsets of a vertex set. A proper vertex colouring has no monochromatic edge. A transversal is a vertex set meeting every edge; its minimum size is the covering number. [S1] [S2]

Statement recorded in the supplied source.

Let $r\geq 3$. For an $r$-uniform hypergraph $G$ let $\tau(G)$ denote the covering number (or transversal number), the minimum size of a set of vertices which includes at least one from each edge in $G$. Determine the best possible $t$ such that, if $G$ is an $r$-uniform hypergraph $G$ where every subgraph $G'$ on at most $3r-3$ vertices has $\tau(G')\leq 1$, we have $\tau(G)\leq t$. [S1]

Residual task reported by the archive.

The embedded review (2026-08-17) records: Determine the sharp t(r). [S1]

Short English statement

For a uniform hypergraph, how large can its covering number be when every sufficiently small subhypergraph has all edges through a single common vertex?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2214 (EP-616), 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 616, https://www.erdosproblems.com/616. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 Erdős, Paul and Hajnal, András and Tuza, Zsolt, Local constraints ensuring small representing sets. J. Combin. Theory Ser. A (1991), 78-84. Bibliographic information from the supplied record.