**Problem**For every , is there a (least) positive integer so that whenever a tournament has its edges colored with colors, there exists a set of at most vertices so that every vertex has a monochromatic path to some point in ?

This problem first appears in print in a lovely paper of Sands, Sauer, and Woodrow [SSW], where they prove that . Actually, they prove a much stronger theorem. They show that for every 2-edge-colored digraph (even infinite) there exists an independent set so that every vertex has a monochromatic path to some point in .

It is open whether is finite for every . The smallest open case, , is already quite interesting; In [SSW] the authors ask if .

## Bibliography

*[SSW] B. Sands, N. Sauer, R. Woodrow, On monochromatic paths in edge-coloured digraphs. J. Combin. Theory Ser. B 33 (1982), no. 3, 271--275. MathSciNet.

* indicates original appearance(s) of problem.