Putting the Combinatorics in Combinatorial Game Theory

Under Conway's simple but powerful game theory axioms, games form a
group with a partial order.  While a great deal has been known about
the group structure of large subsets of games, surprisingly little was
known about the overall partial order.  We prove that games lasting a
fixed number of turns form a distributive lattice, but that the
collection of all finite games does not form a lattice.  We are also
able to give stronger bounds on the number of games born on day n than
those known previously.

We will also present theorems about the structure of this lattice.  A
direct corollary of these theorems is that all maximal chains in the
day n lattice are of the same length, that length being exactly one
plus twice the number of games born by day n-1.

This talk will be introductory and I'll explain the relevant game
theory and lattice theory as we go.

(Work with Dan Calistrate, Bill Fraser, Susan Hirshberg and Marc Paulhus.)