Shattering k-sets with Permutations

Many concepts from extremal set theory have analogues for families of permutations. This paper is concerned with the notion of shattering for permutations. A family P of permutations of an n -element set X shatters a k -set from X if it appears in each of the k ! possible orders in some permutation in P . The smallest family P which shatters every k -subset of X is know to have size Θ ( log n ) . Our aim is to introduce and study two natural partial versions of this shattering problem. Our first main result concerns the case where our family must contain only t out of k ! of the possible orders. When k = 3 we show that there are three distinct regimes depending on t : constant, Θ ( log log n ) , Θ ( log n ) . We also show that for larger k these same regimes exist although they may not cover all values of t . Our second direction concerns the problem of determining the largest number of k -sets that can be totally shattered by a family with given size. We show that for any n , a family of 6 permutations is enough to shatter a proportion between 17 42 and 11 14 of all triples.