This is actually an explanation of birthday paradox.
If you pick random elements out of N, by the time you have picked √N elements you have a good probability of having picked some element twice. You cannot do better than that whatever hash function you use (no hash function can have better collision resistance than uniform random distribution).
You can do worse than that if your hash function doesn’t give uniform distribution on your data, in particular if your input data is manipulated by an adversary. For cryptographically secure hash functions such manipulation is believed (although not proven) very hard.
This is actually an explanation of birthday paradox.
If you pick random elements out of N, by the time you have picked √N elements you have a good probability of having picked some element twice. You cannot do better than that whatever hash function you use (no hash function can have better collision resistance than uniform random distribution).
You can do worse than that if your hash function doesn’t give uniform distribution on your data, in particular if your input data is manipulated by an adversary. For cryptographically secure hash functions such manipulation is believed (although not proven) very hard.