Reu 2007

Bn : (∀x1, . . . , xn)( if all xi are distinct then the subgraph induced on x1, . . . , xn is 3-colorable). By Erdős-DeBruijn, the countable set of axioms Bn (n = 1, 2, . . . ) defines 3-colorability. To show that 3-colorability is not finitely axiomatizable, we show the (apparently) stronger result that non-3-colorability is not axiomatizable. To do this, we use ultraproducts. It suffices to construct graphs Ai which are not three-colorable whose ultraproduct ∏ Ai/μ is three-colorable. Now, to do this, we could find Ai such that girth(Ai)→∞. That would mean that the ultraproduct is cycle-free, i. e., a forest and therefore it is 2-colorable. This proof uses Erdős’s result that there exist finite graphs of arbitrary large girth and chromatic number. The proof was one of the great successes of the Probabilistic Method. We can avoid using this difficult result. Indeed, it suffices to refer to the simpler result, proved in class, that there exist graphs with arbitrarily large oddgirth (oddgirth = the length of the shortest odd cycle) and large chromatic number, because then the ultraproduct will still be bipartite (it will have no odd cycles). Recall that graphs of large oddgirth and large chromatic number can be constructed by iterating the directed line graph construction of §3.1.