This is a good as I could do with MySQL 4.1.10

CREATE TEMPORARY TABLE Temp1
select a.ring_event_id, a.event_name,
       a.ring_id, a.start_time,
       ADDTIME(a.start_time, time_per_dog 
               * count(b.dog_id)) as end_time,
       count(b.dog_id) as num_dogs
 from ring_event a, tda b
 where a.ring_event_id = b.ring_event_id
 group by a.ring_event_id;

CREATE TEMPORARY TABLE Temp2
select a.ring_event_id, a.event_name,
       a.ring_id, a.start_time,
       ADDTIME(a.start_time, time_per_dog 
               * count(b.dog_id)) as end_time,
       count(b.dog_id) as num_dogs
 from ring_event a, tda b
 where a.ring_event_id = b.ring_event_id
 group by a.ring_event_id;

select ring_event_id, person.name, group_concat(skill.skill) as skills
  from person
  join skill_entry on person.person_id = skill_entry.person_id
  join skill on skill_entry.skill_id = skill.skill_id
  join ring_event
 where person.person_id not in (
  select person.person_id
    from tda a
     join dog on a.dog_id = dog.dog_id
     join person on dog.person_id = person.person_id
     join Temp1 b
       on a.ring_event_id = b.ring_event_id
     join Temp2 c
       on (b.start_time < c.end_time and b.end_time > c.start_time)
    where b.ring_event_id = ring_event.ring_event_id )
 group by ring_event_id, name
 order by ring_event_id, name;

drop table Temp1;
drop table Temp2;
[download]

The subselect is getting all people that are busy while ring_event.ring_event_id is in progress. And then the main select gets all the people that aren't busy.

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