Erdős Problem #1174

Does there exist a graph $G$ with no $K_4$ such that every edge colouring of $G$ with countably many colours contains a monochromatic $K_3$?

Does there exist a graph $G$ with no $K_{\aleph_1}$ such that every edge colouring of $G$ with countably many colours contains a monochromatic $K_{\aleph_0}$?

#1174: [Va99,7.91]

set theory | ramsey theory

Disclaimer: The open status of this problem reflects the current belief of the owner of this website. There may be literature on this problem that I am unaware of, which may partially or completely solve the stated problem. Please do your own literature search before expending significant effort on solving this problem. If you find any relevant literature not mentioned here, please add this in a comment.

A problem of Erdős and Hajnal. Shelah proved that a graph with either property can consistently exist.

View the LaTeX source

External data from the database - you can help update this
Formalised statement? No (Create a formalisation here)

1 comment on this problem

Likes this problem	None
Interested in collaborating	None
Currently working on this problem	None
This problem looks difficult	None
This problem looks tractable	None
The results on this problem could be formalisable	None
I am working on formalising the results on this problem	None

When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:

T. F. Bloom, Erdős Problem #1174, https://www.erdosproblems.com/1174, accessed 2026-03-01