On the minimum number of completely 3-scrambling permutations

Abstract

A family P = { π 1 , … , π q } of permutations of [ n ] = { 1 , … , n } is completely k-scrambling [Spencer, Acta Math Hungar 72; Füredi, Random Struct Algor 96] if for any distinct k points x 1 , … , x k ∈ [ n ] , permutations π i 's in P produce all k ! possible orders on π i ( x 1 ) , … , π i ( x k ) . Let N * ( n , k ) be the minimum size of such a family. This paper focuses on the case k = 3 . By a simple explicit construction, we show the following upper bound, which we express together with the lower bound due to Füredi for comparison. 2 log 2 e log 2 n ⩽ N * ( n , 3 ) ⩽ 2 log 2 n + ( 1 + o ( 1 ) ) log 2 log 2 n . We also prove the existence of lim n → ∞ N * ( n , 3 ) / log 2 n = c 3 . Determining the value c 3 and proving the existence of lim n → ∞ N * ( n , k ) / log 2 n = c k for k ⩾ 4 remain open.