Abstract
A family of permutations $F \subseteq S_n$ with a probability distribution on it is called k-restricted min-wise independent if we have $Pr[min \pi(X) = \pi(x)] = 1/|X|$ for every subset $X \subseteq [n]$ with $|X|l.e. k$, every $x\in X$, and $\pi\in F$ chosen at random. We present a simple proof of a result of Norin: every such family has size at least $\binom{n-1}{\lceil k-1/2\rceil}$.
The best available upper bound for the size of such family is $1 + \Sum_{j=2}^k (j - 1)\binom{n}{j}$. We show that this bound is tight if the goal is to imitate not the uniform distribution on $S_n$, but a distribution given by assigning suitable priorities to the elements of $[n]$.
We also investigate the cases where the min-wise independence condition is required only for sets $X$ of size exactly $k$ (where we have only an $\Omega(log log n + k)$ lower bound), or for sets of size $k$ and $k - 1$ (where we already obtain a lower bound of $n - k + 2$).