Erdős Problem #805

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.

For which functions $g(n)$ with $n>g(n)\geq (\log n)^2$ is there a graph on $n$ vertices in which every induced subgraph on $g(n)$ vertices contains a clique of size $\geq \log n$ and an independent set of size $\geq \log n$? In particular, is there such a graph for $g(n)=(\log n)^3$? [S1]

Residual task reported by the archive.

The embedded review (2026-08-17) records: Decide existence at g(n)=(log n)^3 and, more generally, locate the transition between the published lower obstruction and upper construction. [S1]

Short English statement

How small can the tested subsets be while every one of them is guaranteed to contain both a logarithmically large clique and independent set?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2313 (EP-805), 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 805, https://www.erdosproblems.com/805. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 Noga Alon and Benny Sudakov, On graphs with subgraphs having large independence numbers, Journal of Graph Theory 56 (2007), 149-157,

  \nolinkurl{doi:10.1002/jgt.20264.}

  https://arxiv.org/abs/0706.4099. Bibliographic information from the supplied record.

• S4 Noga Alon, Matija Bucić, and Benny Sudakov, Large cliques and independent sets all over the place, Proceedings of the AMS 149 (2021), 3145-3157,

  \nolinkurl{doi:10.1090/proc/15323.}

  https://arxiv.org/abs/2004.04718. Bibliographic information from the supplied record.