Erdős Problem #165

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.

$R(G,H)$ is the least order of a complete graph whose every red--blue edge colouring has a red copy of $G$ or a blue copy of $H$. For integer arguments, $R(s,t)=R(K_s,K_t)$. Write $R(G)=R(G,G)$ and $R(k)=R(K_k,K_k)$. The notation $R(G;q)$ instead uses $q$ colours and the same target in each colour. [S1] [S2]

Statement recorded in the supplied source.

Give an asymptotic formula for $R(3,k)$. [S1]

Short English statement

What is the asymptotic formula, including its leading factor, for the threshold forcing a red triangle or a blue clique of increasing order?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 1969 (EP-165), 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 165, https://www.erdosproblems.com/165. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 Ajtai, Miklós and Komlós, János and Szemerédi, Endre, A note on Ramsey numbers. J. Combin. Theory Ser. A (1980), 354-360. Bibliographic information from the supplied record.

• S4 Bohman, Tom and Keevash, Peter, Dynamic concentration of the triangle-free process. Random Structures Algorithms (2021), 221-293. Bibliographic information from the supplied record.

• S5 M. Campos, M. Jenssen, M. Michelen, and J. Sahasrabudhe, A new lower bound for the Ramsey numbers $R(3,k)$. arXiv:2505.13371 (2025). Bibliographic information from the supplied record.

• S6 Z. Hefty, P. Horn, D. King, and F. Pfender, Improving $R(3,k)$ in just two bites. arXiv:2510.19718 (2025). Bibliographic information from the supplied record.