TITLE: In Szemerédi's Regularity Lemma, how many blocks are in the partition?
QUESTION [9 upvotes]: Suppose $G = (V,E)$ is a directed graph.
For sets $A$ and $B$ of vertices of $G$,
let $d(A,B) = |(A \times B) \cap E| / (|A||B|)$ denote the edge density between $A$ and $B$,
and say that the pair $A,B$ is $\epsilon$-regular if
$$ |d(X,Y) - d(A,B)| \lt \epsilon $$
whenever $X \subseteq A, Y \subseteq B$, $X$ contains more than an $\epsilon$-fraction of the vertices of $A$, and $Y$ contains more than an $\epsilon$-fraction of the vertices of $B$.
An equipartition is a partition with the sizes of blocks of the partition differing pairwise by at most 1.
Szemerédi's Regularity Lemma can then be stated as:

For any $\epsilon > 0$, there exists $M(\epsilon)$ such that for every graph $G=(V,E)$ there is some $k\le M(\epsilon)$ and an equipartition $V = V_1 \cup \ldots \cup V_k$ in which each block $V_i$ contains at most
  $\lceil \epsilon |V|\rceil$ vertices,
  and having the property that for all but at most $\epsilon k^2$ of the pairs $(i,j)$, the pair $V_i, V_j$ is $\epsilon$-regular.

Do we have any bounds or asymptotics for how $M(\epsilon)$ behaves as a function of $\epsilon$?
I vaguely recall having read a comment that $M(\epsilon)$ is likely to be extremely large, making the Regularity Lemma only useful for truly large graphs.  But I have not been able to find this assessment again, so a pointer would be appreciated.  (I did check Terence Tao's exposition again, and some of the more obvious references.)

REPLY [7 votes]: In addition to Gowers' breakthrough paper, a recent paper addressing the bounds in Szemerédi's regularity lemma, and its weak and strong variants is: 
D. Conlon and J. Fox, Bounds for graph regularity and removal lemmas, Geom. Funct. Anal. 22 (2012), 1191-1256.
It is shown that the bound on the number of parts for Szemer\'edi's regularity lemma grows as at least a tower of $2$'s of height $\Omega(\epsilon^{-1})$, for the Frieze-Kannan weak regularity it grows as $2^{\Theta(\epsilon^{-2})}$, and for the strong regularity lemma of Alon, Fischer, Krivelevich, and Szegedy it grows as a wowzer in a power of $1/\epsilon$. The wowzer function is the next function in the Ackermann hierarchy after the tower function. 
Update (April 2014): Moshkovitz and Shapira recently gave a simple proof of a bound which is a tower of height a power of $1/\epsilon$ in the paper entitled: 
A Short Proof of Gowers' Lower Bound for the Regularity Lemma.
Laszlo Miklos Lovasz and I recently give a bound on the tower height in a version of Szemer\'edi's regularity lemma which is tight up to an absolute constant factor in the paper entitled:
A tight lower bound for Szemerédi's regularity lemma.