4. From $1,2, \cdots, 2006$, at least
$\qquad$ odd numbers must be taken to ensure that there are definitely two numbers whose sum is 2008.
4.503.

From $1,2, \cdots, 2006$, selecting two odd numbers whose sum is 2008, there are a total of 501 pairs as follows:
$$
3+2005,5+2003, \cdots, 1003+1005 \text {. }
$$

Since 1 added to any of these odd numbers will not equal 2008, therefore, at least 503 odd numbers must be selected to ensure that there are definitely two numbers whose sum is 2008.