**Conjecture**For every , there exists an integer such that if is a digraph whose arcs are colored with colors, then has a set which is the union of stables sets so that every vertex has a monochromatic path to some vertex in .

In the particular case of tournaments (and more generally when the stabilty number of is bounded), it has been proved by Bousquet, Lochet, and Thomassé [BLT].

## Bibliography

[BLT] Nicolas Bousquet, William Lochet, Stéphan Thomassé: A proof of the Erdős-Sands-Sauer-Woodrow conjecture,

[SSW] B. Sands, N. Sauer and R. Woodrow, On monochromatic paths in edge-coloured digraphs. Journal of Combinatorial Theory, Series B, 33, (1982), 271--275.

* indicates original appearance(s) of problem.