In my textbook, Szemeredi’s regularity lemma states as
$\forall\epsilon>0,\forall$ integer $m$ $\exists$ integer $M$ $\forall$ graph $G$ on $n\ge m$ vertices $\exists k$ with $m\le k\le M$ and an $\epsilon-$regular partition $(V_0,V_1,V_2,\cdots,V_k)$ of $V(G)$.
But in some other textbook, it states that:
For every $\epsilon > 0$ and every integer $m$ there exists an integer $M$ such that for every graph
$G$ on $n \ge m$ vertices there exists an integer $k$ such that $m \le k \le M$ and $V (G)$ can be
partitioned into $(V_1, V_2, . . . , V_k)$ in such a way that $||V_i
| − |V_j|| \le 1$ for every two distinct integers $i, j \in \{1, 2, . . . , k\}$ and all but $\epsilon k^2$ of the pairs $(V_i, V_j )$ are $\epsilon$-regular.
We say that $(A,B)$ is $\epsilon-$regular in $G$ if for all $X\subset A,Y\subset B$ with $|X|\ge\epsilon|A|$ and $|Y|\ge\epsilon |B|$ we have $|\frac{|<A,B>|}{|A||B|}-\frac{|<X,Y>|}{|X||Y|}|\le\epsilon$, and $<C,D>=\{e:e \text{ has one end in $C$, the other in $D$}\}.$
A partition $(V_0,V_1,\cdots,V_k)$ of $V(G)$ is $\epsilon-$regular if
(1) $|V_0|\le\epsilon |V(G)|$
(2) $|V_0|=|V_1|=\cdots=|V_k|$
(3) all but at most $\epsilon k^2$ pairs $(V_i,V_j)$ are $\epsilon-$regular $(1\le i,j\le k)$.
I didn't read through the proof to Szemeredi's Lemma. I'd like to know wether I could deduce one from the other. For example, if we want to get the second version, it seems that we need to assign the vertices in $V_0$ to other parts evenly and keep the $\epsilon-$regular property. But I'm not sure how to show it strictly.