As in the title, I am trying to find the largest (aka least upper bound) of a (very large) set of integers. Importantly, I do not have direct access to the full list of integers, but I do have a function $f(n)$ which returns true/false if $n$ is in the set. The function $f(n)$ is expensive and I would like to minimize the number of calls I must make to it.
The integers might or might not be consecutive, or have large gaps between them (i.e. might be sparse or dense). There is no prior-known upper bound on the largest integer in the set, which can go off to infinity in theory.
Is there a well-trodden algorithm for doing this? My inkling is to do some kind of random sample to determine the density, and then try to find the upper bound within some certainty. I'm not sure how to bound my initial sample properly then though, or which distribution I might assume the integers have based on that sample.