There are different answers depending on the end goal.

Mere survival: Isolated human populations have been bottlenecked to as few as a few hundred individuals and survived, IIRC.

A quick search says biologists like to see 25+ breeding pairs to maintain an animal species (if I'm reading that correctly). So 50-100 seems like pretty close to the minimum.

Long-term colony building with full genetic diversity needs a lot more: At least one estimate is as high as 40,000 people. The high number is for Earth-like diversity in the population, and with no need for any overarching breeding program, so it's really kind of an outlier scenario. That 40k figure can be pared down significantly if you have strict protocols, or accept some loss of diversity.

So anywhere from 50 people to 40,000 people, but the end result will look wildly different at the extremes.

Pro tip: take frozen sperm/eggs with the ship. Can be used to infuse the population with fresh genetic material.

Around 4000.

When inbreeding effects are included, estimates of MVP for many species are in the thousands. Based on a meta-analysis of reported values in the literature for many species, Traill et al. reported concerning vertebrates "a cross-species frequency distribution of MVP with a median of 4169 individuals (95% CI = 3577–5129)."[8]

This has been modeled. Though I don't have any references at hand, this is what I remember.

If you want to allow people to choose mates and breed normally, you'd need at least 3000 people, 4 to 5000 would be better.

If you are strictly controlling genomes and breeding pairs, ignoring monogamy and social norms. You might be able to get away with 100 if you selected for maximum initial genetic diversity. But 200 would be easier.

The obvious solution here is to include frozen gametes in the ship’s cargo to increase the diversity of the population. It would be culturally easiest with frozen sperm but if necessary you could include eggs as well, provided people are willing to be surrogate mothers.

It's the wrong place to ask, but if I liked this question, where should I be subscribed to besides !space@mander.xyz ?

In the Galaxy's Edge series this is tangentially addressed, the TLDR is that there are other implications of being on a ship for generations that outweigh the genetic problems. The Savages have a well earned title fwiw both in origin and action. It is a bleak outlook on the concept but I don't think it is farfetched given everything we are living through right now.

There are actually two issues:

• The most obvious effect of inbreeding is the increase in homozygosity for deleterious mutations, causing more birth defects.

• A subtler effect is the loss of genetic diversity reducing a population’s ability to continue to evolve in response to future selection pressures. This would be especially important when migrating to a new environment with new selection pressures the species has never encountered before.

Not what you mean, but one woman and a shit-ton of frozen sperm and ova would probably work, say 3 for redundancy...