Query with column data for a max

In Postgres, I have a table like this:

    Column    |            Type             |                                 Modifiers
--------------+-----------------------------+--------------------------------------------
 problem_pk   | bigint                      | not null default nextval(...)
 version      | bigint                      | not null default 1
 problem_type | e_problem_type              | not null
 building_fk  | bigint                      | not null
 dpoint_fk    | bigint                      |
 native_name  | character varying(256)      |
 event_date   | timestamp without time zone | not null default now()
 count        | bigint                      |

and e_problem_type looks like this:

\dT+ e_problem_type;
                                         List of data types
  Schema  |      Name      | Internal name  | Size |             Elements             | Description
----------+----------------+----------------+------+----------------------------------+-------------
 ecorithm | e_problem_type | e_problem_type | 4    | nil                             +|
          |                |                |      | empty_string                    +|
          |                |                |      | nan                             +|
          |                |                |      | dash                            +|
          |                |                |      | other_bad_data                  +|
          |                |                |      | unmapped                        +|
          |                |                |      | missing_header                  +|
          |                |                |      | low_data_volume                 +|
          |                |                |      | attachment_with_no_valid_headers |


And I have a query like this:

select
    problem_type, native_name, max(event_date)
from
    point_problem_history
where
    building_fk = 3
and
    problem_type = 'nil'
group by
    native_name, problem_type
order by native_name;

I want to add the "count" field to my output, but have been unsuccessful in figuring out how to do this. If I add it, I have to group by it and I end up with duplicate native names. What I want is, for each native name, the value of the 'count' field for the most recent event_date for that native name and problem type = (the string 'nil'). I've tried a lot of different queries and subqueries with no success. Any and all help appreciated.