Can you cross 3 different roosters with a single hen and then breed the offspring between them?

[BigBlueHen53]

Yes you breed each rooster in turn to the hen, making sure to keep a time period between so there is no question who the father is, and keeping a record of each batch.
Then once the chickens have grown up you cross the half sibs.
Uncles and neices, Aunts and nephews also work in this type of line breeding.
Far too many humans worry about line breeding because they do not understand it.
1. If you have healthy genes only, you can get healthy genes only.
2. If you have unhealthy genes then you will get unhealthy genes.
3. If the birds are related any unhealthy genes will show up faster.
4. If the birds are not related it is no guarantee of healthy genes.
5. If you cross 2 related birds with only healthy genes you can do this for several generations before needing an outcross.

I have been doing this with finches for the past close on 40yrs and it is rare for anything funky to happen.
The problem is that the royal families of Europe inbred with funky genes and so did the royal families of Egypt. So those 2 examples have tainted peoples perceptions.

Instead look at the success stories of endangered species bought back from the brink with healthy populations of less than 500 e.g. the Kakapo and Hyacynith Macaw

Thank you for the simple clarification.