Viral load. Number of viral copies per volume of fluid.
Merely exposing someone to a virus does not necessarily result in widespread infection. A low enough number of copies could be neutralized by the host's immune response before it has a chance to spread significantly. How low this number is depends on the virus. For example, COVID-19 load is higher in symptomatic patients.
Chronic viral infections are managed by reducing as much as possible the number of viral copies in circulation. Risk of transmission is mitigated when number of copies is low enough. There are studies showing risk of HIV transmission approaches zero when HIV load < 200 copies/ml. Ideal would be HIV load < 50 copies/ml or undetectable.
At that low virion count it’s more about the probability that a virion will attach to the correct CD4 receptor and be able to work inside the cell. It’s not about the host's immune system defeating or being defeated on a number basis.
There is an "independent action hypothesis" in virology. It states that each virus particle (virion) has an independent probability of starting the infection in the body.
So it it turns out true, there is no "safe level" - we could talk about an extremely small probability, but it would still be higher than 0 - not exactly a risk one would like to take.
Yes. Assume 1 virion has a 0.001 chance to start an infection. You could infect 1 person with 692 virions, resulting in a 50% change of them getting ill (1-0.999^692). Or you could infect 692 people each with 1 virion, with 50% chance of at least 1 of them getting ill (35% chance: 1 ill, 12% chance: 2 ill, 3% chance: 3 ill, ...).