oh there are many different solutions, i just mentioned a few of the more realistic ones. materials science will need to improve by several magnitudes before true arcologies are possible. but yes, it's unlikely that we'll develop the tech we need in the timeframe.

carrying capacity? i looked into it several years ago after reading about it somewhere. probably sci-fi, I do enjoy those. on the low end, it's around 2 billion. an 80% population drop may be required. there will be resource wars before that happens though & quite likely a few exchanges of strategic nuclear weapons. presumably the famines would drastically increase after that? I supose we'll see.