[...] what is the idea now then? I have introduced a frame of eggs into a brood box that is being filled with honey... if they make a queen, she hatches, mates, and returns to this hive won't they swarm straight away as they have no space? What is my recourse here?
Phil - I understand your frustration. I've been watching this thread and haven't commented thus far, as honey flows such as this are completely out of my experience. What I'm about to suggest is simply what I imagine I'd do if finding myself in your situation, and so if anyone with actual experience of such events wants to contradict these suggestions - that would be highly desirable.
As I see it, you need to take pressure off the brood nest area. I'd do this in two ways: firstly, by removing the amount of forager traffic. I don't know how many colonies you have, but if there's one with little forager activity (can be difficult to judge, I know), then swap-over your problem hive with that one. If they're all highly active, then consider moving your problem hive well away into a new location (i.e. far enough away so that the foragers won't return to it), and if necessary, adjust the other hive positions so that the 'lost foragers' will be easily able to find themselves a new home. I say this on the assumption that the loss of your 'problem colony' would be more important to you than a reduction in honey crop from it. If there ARE any swarm cells present, make the move before a virgin emerges. You'd be very unlucky indeed if there happened to be a virgin present and awaiting her nuptial flight, the success of which a move would of course sabotage.
Ok - secondly, I'd do exactly what Jim suggested in the first reply - add an empty super to that hive (with pre-drawn combs if possible) - and, on the assumption that bees would much rather store honey above the brood nest than in it, I would expect them to move the existing nectar upwards and thus clear the brood nest area for laying.
Then - it's just a waiting game. By (hopefully) clearing the brood area, you will have done the best you can, and from then on it's down to the bees. Denying that colony significant input from the field-foragers, AND by giving them empty space in which to store whatever honey has already been collected, will hopefully be enough to convince them that life isn't quite as rosy as it was yesterday, and so any thought of swarming right now probably isn't such a smart idea ...
Putting in the frame of eggs/larvae was a good insurance move - hopefully it won't be needed.
There is one other thought I've had, and that is to create an additional entrance directly to the supers during such a major flow - this could easily be done by offsetting a super, or the crown board (inner cover) from the box below it by just enough to create a gap of say, 8mm, to give the foragers direct access into that super. Easy enough to then seal that access by replacing the boxes properly when the flow subsides.
Hopefully there's an idea or two in the above of some use.