Avoiding monochromatic maximal antichains

Abstract

A vertex coloring of a (possibly infinite) poset P 1s called good iff it leaves no nontrivial maximal antichain in P monochromatic. What is the minimum number of colors for which P admits a good coloring? By extending the result for finite posets, it can be shown that if P is well-founded and contains an element with no maximal antichain above it, then P admits a good three-coloring. For products of chains we exploit properties of cofinal and coinitial sequences to obtain good two-colorings in certain cases, the covering chain and club coloring results. As well, we introduce the concept of half-maximal antichain for its potential applications and its own merit. While attempting to extend the positive results thus far obtained, we found examples that violated the conditions of those results. At this point we are unable to determine the number of colors required by such examples.