Min-wise independent groups (Q1413231)

Let \(\Omega\) be a finite set of cardinality \(n\) with a linear order on it and let \(k\) be a positive integer. Let \(F\) be a set of permutations on \(\Omega\) and let \(\Pr_{\mathcal F}\) be an arbitrary distribution of probability on \(\mathcal F\). The set \(\mathcal F\) is said to be biased \(k\)-restricted min-wise independent if for every subset \(X\) of \(\Omega\) such that \(|X|\leq k\), and every \(x\in X\), when \(\pi\) is chosen at random in \(\mathcal F\), we have that \(\Pr_\mathcal F(\min \pi(X))=\pi=1/|X|\). The authors study biased \(k\)-restricted min-wise independent groups. In particular, they prove that when \(k\) is close to \(n\), then biased \(k\)-restricted min-wise independent groups are transitive.