Multiple twins in permutations

By an r-tuplet in a permutation we mean a family of r pairwise disjoint subsequences with the same relative order. The length of an r-tuplet is defined as the length of any single subsequence in the family. Let tprq(n) denote the largest k such that every permutation of length n contains an ( ) r-tuplet of length k. We prove that tprq (n) = O nr 2r-1and tprq(n) = ( ) omega nR 2R-1 , where R = (2r ' 1). We conjecture that the upper bound brings r the correct order of magnitude of tprq(n) and support this conjecture by proving that it holds for almost all permutations. Our work generalizes previous studies of the case r = 2.