Known to be at most 5, witnessed by the following path:
Erdös Fan Rong K. Chung / Stephen T. Hedetniemi / Maria Margaret Klawe Robert E. Tarjan Bob Paige Fritz Henglein Christian Stefansen
and also by
Erdös Ronald L. Graham Jeffrey D. Ullman Harry G. Mairson Fritz Henglein Christian Stefansen
Slightly modified version of the paths found on Henning Makholm's homepage.
Read about The Erdös Number Project.