Coloring Ordered Sets to Avoid Monochromatic Maximal Chains

This paper is devoted to settling the following problem on (infinite, partially) ordered sets: Is there always a partition (2-coloring) of an ordered set X so that all nontrivial maximal chains ofX meet both classes (receive both colors)? We show this is true for all countable ordered sets and provide counterexamples of cardinality H3. Variants of the problem are also considered and open problems specified. 0. Introduction. It is obvious that if a finite ordered set contains no one-element maximal chains (isolated points) then the set can be 2-colored so that every maximal chain receives both colors—let the maximal elements be blue and the rest, red. More generally, the top half of a finite ordered set can be made blue, and the bottom, red: every maximal chain makes an appearance in each half. But in the infinite case, can we halve in a similar manner and guarantee that every maximal chain intersects both halves? QUESTION 1. Given an ordered set X without isolated points, is there a 2-coloring of the elements of X by blue and red so that each maximal chain receives both colors and so that the blue set is a final segment and the red, an initial segment of X? In Section 1 we show that the answer is yes for countable orders but that in general, there fails to be such a partition. We can ask for somewhat less. QUESTION 2. Given X, without isolated points, is there a 2-coloring so that each maximal chain receives both colors? In Section 2 we consider examples of partial orders for which the answer to Question 2 is positive, including the counterexamples to the first question. For instance a finite product of scattered chains admits a good 2-coloring. It is not the case that all scattered orders have good 2-colorings (Example 2 of Section 4 settles this). However, we do not know whether all finite products of arbitrary chains admit such colorings. Section 3 is comprised of some small examples (of size the continuum) showing that, for them, it is at least consistent that good 2-colorings always exist. In Section 4, we settle Question 2 in the negative by providing examples of orders for which all 2-colorings have monochromatic maximal chains. Indeed we show that for The work of the first author was supported by ONR Contract N00014-K-85-0769. The work of the third author was supported by NSERC Grant 69-1325. The work of the fourth author was supported by NSERC Grant 69-0259. Received by the editors July 5, 1990 . AMS subject classification: 06A10.