Some interesting problems remain, for instance:TheoremIt is NP-complete to determine if a given ordered set has an order-preserving fixed-point-free self-map.

Is there a natural self-computation for FPFMAP?Applying a graph theoretic result [A. Lubiw 1981], it can be shown, even for length one, that it is NP-complete to determine if a given ordered set has a fixed-point free

Are two given order-preserving maps homotopic?I will continue investigation into the above problems using constructive methods and the current interactive proof techniques.

Given an ordered set, how many colors are needed so that each maximal chain (or antichain) receives at least two?Early results [Duffus, Rödl, Sauer, and Woodrow, 1992] used an Erdös-Rado Ramsey theorem, thereby producing rather large examples. We have made considerable progress in finding smaller and more natural examples by exploiting properties of cartesian products of chains with gaps [D. Duffus, T. Goddard 1995 (accepted), 1996 (preprint)]:

Moreover, these techniques can be applied to the dual question to give the first results for 2-coloring maximal antichains in infinite ordered-sets:TheoremThere is a two-dimensional ordered set of cardinality $\aleph_1$ for which every subset or complement thereof contains a maximal chain. This holds even in ZFC.

Still, much remains; for instance, is there an ordered set of size $\aleph_1$ for which every finite coloring leaves monochromatic maximal chains?TheoremThere is a two-dimensional ordered set of cardinality $2^{2^{\aleph_0}}$ that must be colored with infinitely many colors before no monochromatic maximal antichains are present.

Given an ordered set with no infinite antichain, does there exist an antichain partitition and a chain so that the chain meets each member of the partition?This problem is believed to be very challenging; our positive result holds in a rather special situation.

I will continue by seeking results in other special cases, as the full problem has a strong connection with the König Duality theorem (so called by Aharoni and Korman), a deep result which is the product of a substantial body of work.TheoremIf $C$ is any chain and $P$ is any finite ordered set, then $C \times P$ can be partitioned into antichains in such a way that there is a common chain meeting each of them.

How can networked computers improve mathematics education?The World Wide Web is rapidly becoming a fixture in the academic and commercial spheres, however few resources exist to support the mathematics education of secondary and college students. We must experiment with hypertext and electronic interaction to see how they can enhance learning.

Currently [ http://sirius.mathcs.emory.edu/~goddard/math112] we have a message board for the class, along with assignment updates, the class syllabus, and even final grades. The system is very flexible. The main problem lies with making it approachable for students with little or no experience with computers; this will require progressive refinement.