store posts_id of category into a varable?

For getting the posts related to each category you need to use get_posts( $args ) function and in $args you can add your category_id

$args = array ( 
    'post_type'   => 'post',    
    'category'    => $cat_ID,
    'post_status' => 'publish',
);

$my_posts = get_posts( $args ); // return array of posts

// storing all post ID in an array
foreach( $my_posts as $my_post ) {
    $post_cat_name[] = $my_post['ID'];
}

So in $post_cat_name it will store all the post id related to your $cat_ID

If you want these to be done for 3 $cat_id, then you can loop the above code for 3 $cat_id and you can use $post_cat_name array as 2D array.