TITLE: Counterintuitive consequences of the Axiom of Determinacy?
QUESTION [31 upvotes]: I just read Dr Strangechoice's explanation that if all subsets of the real numbers are Lebesgue measurable, then you can partition $2^\omega$ into more than $2^\omega$ many pairwise disjoint nonempty subsets.  Note that this paradoxical conclusion is a consequence of the Axiom of Determinacy.  
I've read of many counterintuitive consequences of the Axiom of Choice, but what other consequences come from the Axiom of Determinacy?  
(I am aware that AD is inconsistent with AC, and yes, AC being false is rather counterintuitive.)

REPLY [17 votes]: Graph combinatorics is very different under AD to what we are used to under AC. For example, under AD there is an acyclic graph on $\mathbb{R}$ with no 2-coloring (indeed, no coloring with countably many colors). Hall's marriage theorem is also false, even in a very trivial case: there is a 2-regular bipartite graph on $\mathbb{R}$ with no perfect matching.
These theorems both follow from the assertion that every set of reals has the Baire property, and the arguments can be found in Kechris-Solecki-Todorcevic's 1999 paper: Borel Chromatic Numbers.
I also have a recent paper on descriptive combinatorics that directly uses the axiom of determinacy, and not just measure or category arguments. For example, I show that under AD, for every n, there is an acyclic n-regular graph on $\mathbb{R}$ with chromatic number n+1, and an acyclic bipartite n-regular graph with no perfect matching. The games I use are based on partitions of free products of countable groups into two pieces.