Unions of triangle free graphs

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A graph is $K_4$-free if no four vertices are pairwise adjacent. A spanning subgraph is triangle-free if it contains no three pairwise adjacent vertices. Here $\aleph_0$ is the cardinality of the natural numbers; graph cardinalities are unrestricted.

Statement recorded in the supplied source.

Is there a $K_4$-free graph $G=(V,E)$ such that no sequence of triangle-free spanning subgraphs $(V,E_1),(V,E_2),\ldots$ satisfies $E=\bigcup_{j\geq1}E_j$? Equivalently, is there such a graph with no countable edge-colouring in which every triangle uses at least two colours? [S2]

Short English statement

Does some graph without a four-clique resist every countable cover of its edges by triangle-free subgraphs?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3298 (OPG-349). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Unions of triangle free graphs, node 349, original problem page. Original question and full discussion. https://www.openproblemgarden.org/op/unions_of_triangle_free_graphs.