Erdős Problem #159

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.

$K_t$ is a complete graph and $C_t$ a simple cycle of length $t$. A complete multipartite graph has no edges within parts and all edges between different parts. An independent set has no internal edge; a clique has all its possible internal edges. [S1] [S2]

Statement recorded in the supplied source.

There exists some constant $c>0$ such that \[R(C_4,K_n) \ll n^{2-c}.\] [S1]

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

Short English statement

Can the Ramsey threshold for a red four-cycle or blue large clique be bounded by a fixed power strictly below two?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 1965 (EP-159), 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 159, https://www.erdosproblems.com/159. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 Paul Erdos, R. J. Faudree, C. C. Rousseau, and R. H. Schelp, On cycle-complete graph Ramsey numbers, Journal of Graph Theory 2 (1978), 53-64,

  \nolinkurl{doi:10.1002/jgt.3190020107.}

  https://doi.org/10.1002/jgt.3190020107. Bibliographic information from the supplied record.

• S4 Joel Spencer, Asymptotic lower bounds for Ramsey functions, Discrete Mathematics 20 (1977), 69-76,

  \nolinkurl{doi:10.1016/0012-365X}

  (77)90044-9. https://doi.org/10.1016/0012-365X(77)90044-9. Bibliographic information from the supplied record.

• S5 Erdős, Paul and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., On cycle-complete graph Ramsey numbers. J. Graph Theory (1978), 53-64. Bibliographic information from the supplied record.