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I don't know the exact equasions behind those numbers, I just know, that this was calculated and such-and-such numbers are used.
So a first estimate that 2 completely unrelated mating pairs from distant parts of the human genetic tree might make a minimal gene pool candidate. Actually it is a very important question for laboratory animals (and, I imagine, endangered species) and was calculated to be 25 couples.
It's not like you're controlling who is in love with who or who is having sex with who.
nytimes.com/2011/11/27/magazine/…I am no expert on dogs, but did learn a bit about in- and outreeding. In a closed breed, only offspring of animals already belonging to this race belong to the race. and mate them, the offspring is about 0,5-1% more inbred than their parents (meaning both their copies of any gene are a tiny bit more likely to be inherited from one ancestor mouse).
In an open breed any animal that looks similar enough (or has other important traits, characteristic for the breed) can be a parent of a "breed" litter. too few animals in the breed, too bad overall health of the breed) a closed breed can be opened and after the problem is solved, closed again. If you have enough animals, this effect is smaller than the effect of other phenomena, such as genetic drift (compare with the assumptions behind Hardy-Weinberg equilibrium).
With any number of animals (including humans), there is always some inbreeding happening, but you can reduce it with the number of breeding pairs and careful pairing.When you get to 25 pairs (50 animals) and have complete control over pairing, you can sustain the genetic diversity practically infinitely (especially if you take into account spontaneous mutations).