Abstract :
We say that an ordered set has a k-coloring if its elements can be colored with k colors such that no maximal nontrivial antichain is monochromatic. It was shown by Duffus, Kierstead and Trotter that each ordered set has a 3-coloring. Very few examples of ordered sets not admitting 2-colorings have been found so far. The smallest of them has 17 elements. We consider a certain subclass of the class of ordered of width 3 and prove a necessary condition satisfied by ordered sets in this subclass that are not 2-colorable. The condition allows us to find several new examples of ordered sets without a 2-coloring. Moreover we show that every ordered set without a 2-coloring in the considered subclass contains the above mentioned smallest known 17-element ordered set without a 2-coloring.
