The exceptional three-color path problem: star-forest and triangle-density reductions

Definitions and mathematical statement

A graph here is finite, simple, and undirected. A proper vertex coloring assigns different colors to adjacent vertices; the chromatic number $\chi(G)$ is the minimum number of colors permitting such an assignment. An edge coloring with three colors is an arbitrary function $c:E(G)\to\{1,2,3\}$ and is not required to be proper. A path of length three consists of four distinct vertices $v_0,v_1,v_2,v_3$ with edges $v_0v_1,v_1v_2,v_2v_3$. It is monochromatic when these three edges receive the same color. Extra edges between its vertices are permitted.

Gyárfás's question is whether \[ \chi(G)\ge6\quad\Longrightarrow\quad \text{every }c:E(G)\to\{1,2,3\} \text{ has a monochromatic path of length three}. \] The universal quantifiers range over all such graphs and all three-color assignments to their edges. Equivalently, suppose the edge set can be partitioned into three spanning subgraphs, each containing no three-edge path. Must their union be properly vertex colorable using at most five colors? A missing edge belongs to none of the three color classes and cannot be used to form a path. Thus the claim concerns arbitrary graphs of chromatic number at least six, without any completeness or minimum-degree assumption. The path notation in the workshop counts edges, so its $P_3$ is the four-vertex path.

Short English statement

Does every three-coloring of the edges of a graph requiring six vertex colors contain a monochromatic three-edge path?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20000925 (AIM-COMBINATORICS-0050), AIM Workshop Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 American Institute of Mathematics, Graph Ramsey theory, item 3. Original workshop question underlying AIM-COMBINATORICS-0050. https://www.overleaf.com/read/mnvcscjjysvg.