Erdős Problem #600

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.

A graph has a vertex set $V$ and edges that are unordered pairs of distinct vertices. A subgraph need not be induced unless that word is stated. A proper vertex colouring gives adjacent vertices different colours; $\chi(G)$ is the least number of colours. [S1] [S2]

Statement recorded in the supplied source.

Let $e(n,r)$ be minimal such that every graph on $n$ vertices with at least $e(n,r)$ edges, each edge contained in at least one triangle, must have an edge contained in at least $r$ triangles. Let $r\geq 2$. Is it true that $ e(n,r+1)-e(n,r)\to \infty $ as $n\to \infty$? Is it true that $ \frac{e(n,r+1)}{e(n,r)}\to 1 $ as $n\to \infty$? [S1]

Residual task reported by the archive.

The embedded review (2026-08-17) records: Prove either the difference divergence or the ratio limit. [S1]

Short English statement

For dense graphs whose every edge is triangular, do successive book-size thresholds differ unboundedly but have ratio tending to one?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2204 (EP-600), 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 600, https://www.erdosproblems.com/600. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 Ruzsa, I. Z. and Szemerédi, E., Triple systems with no six points carrying three triangles. Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. II (1978), 939-945. Bibliographic information from the supplied record.