2. From $1,2, \cdots, 2008$, at least $\qquad$ even numbers must be taken to ensure that there are definitely two even numbers whose sum is 2012.
2.504.

From $1,2, \cdots, 2008$ select two even numbers, the sum of which is 2012, there are a total of 501 pairs, that is
$$
4+2008,6+2006, \cdots, 1004+1008 \text {. }
$$

Since the sum of 2 or 1006 and any one of the even numbers above does not equal 2012, therefore, at least $501+2+$ $1=504$ even numbers must be taken out to ensure that there are definitely two even numbers whose sum is 2012.