I can vouch for the fact that having 3 or more brood chambers (deep size)
will NOT prevent swarming.  It will, however result in some mighty big
swarms!
 
Something to consider regarding singles vs. doubles is the queen's time
spent searching for empty cells.  When given some nice empty combs, it is
not too surprising that she can fill them so fast.  The search-time then is
very brief, between eggs laid.  Queens in observation hives can be seen to
spend an awfully large amount of time just walking around, checking cells
for vacancies.  It can take a while before she finds an empty cell ready
for an egg.  I wonder how this
variable figures into the common 2000-eggs-a-day estimate (?).
 
This year I am running 18 of my colonies on single brood chambers, each
topped with a queen excluder.  There are 9 combs in each brood chamber, and
they are good quality worker comb, mostly.  This seems to be plenty and a
good size hive to deal with through the summer.  It has proved effective
for me in the past, often with solid brood wood-to-wood and even right
across the hive to the walls.
 
These colonies overwintered in 1 deep plus 1 medium, and they were reversed
(switched positions) last month.  This was to encourage the queen to
establish the nest well in the deep box.  This past week I went out and put
each deep box (incl. queen) on the floor, added an excluder and empty super
(combs not in the frames, just starters).  Then a honey super of combs, and
finally the medium box that had LOTS of brood in it.  I was surprised to
find that only a couple of the colonies were preparing to swarm.  Hopefully
the space right above the brood nest will keep the bees occupied, and the
brood at the top will draw up a lot of bees, easing the congestion
downstairs.  Have to wait and see...  Some of the deep brood chambers were
getting plugged with honey already and were quite heavy.  Now that they are
on the bottom, the bees will move most of the nectar/honey up into the
supers.  At least that's what I told them to do...
 
Fruit bloom pretty much over now, yellow-rocket and tartarian honeysuckle
are very prominent now, keeping the colonies very usy.  Black locust isn't
far away!
 
best wishes
 
JG