The demographics matter a great deal. If the "vaporization" were random you would have major cities with very large populations with the median age being ~30 and most everyone being 55 years old or less. Source
The top 20 cities around the world would have a population of nearly a million with the largest city in China having 3 million. Source
A society needs 6 basic things to become stable, food, water, shelter, safety (defense from predators), healthcare, and importantly - numbers. We need a medium sized group of people to survive and thrive. So very rural areas where only one person in 200 km^2 is not going to last very long and definitely not going to be procreating.
Again, if the event was random then you can expect that in most major cities you would still have people in a variety or professions - farmers (or at least gardeners), healthcare workers, scientists, engineers, teachers, and importantly, leaders. And of course the 'bad' folks who can't focus on the long term vision needed to stabilize a society.
People would have to band together to accomplish the goals above.
They would likely need to move away from the cities to more agricultural areas or places with large quantities of game for food.
Organization would be paramount.
Within the first few years these 'city dwellers' would learn the basics needed for survival with the population decreasing during this learning period. How much it decreased would be entirely based on how they dealt with raiders and anarchists.
Given a decade we would again have some population centers.
That would also be about the time the old stuff would last, gasoline, gunpowder, etc. and a new set of skills would become important - we would again have to learn how to build and maintain our technology. Hopefully the leaders foresaw this and had people finding books to maintain the library of knowledge we've built up over the centuries.
Once they overcome this hurdle, the next 20 years or so life will become more routine and babies will be being born. I suspect if each of these new population centers makes it 50 years they will have stabilized enough for populations to grow.
It is a bit optimistic but I do feel it is also realistic since we have so many major cities in the world. NY would have over 200,000 people left - that's a decent sized city with enough diversity for them to survive and thrive.