# Exact colorings of graphs

**Conjecture**For , let be the statement that given any exact -coloring of the edges of a complete countably infinite graph (that is, a coloring with colors all of which must be used at least once), there exists an exactly -colored countably infinite complete subgraph. Then is true if and only if , , or .

Stacey and Weidl have shown that given , there is an integer such that is false for all .

## Bibliography

* M. Erickson, ``A Conjecture Concerning Ramsey's Theorem,'' *Discrete Mathematics* **126**, 395--398 (1994); MR **95b**:05209

A. Stacey and P. Weidl, "The Existence of Exactly m-Coloured Complete Subgraphs," *J. of Combinatorial Theory*, Series B **75**, 1-18 (1999)

* indicates original appearance(s) of problem.