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The sequence a_1,...,a_m is a common subsequence in the set of permutations S= {p_1,...,p_k} on [n] if it is a subsequence of p_i(1),...,p_i(n) andp_j(1),...,p_j(n) for some distinct p_i, p_j in S. Recently, Beame andHuynh-Ngoc (2008) showed that when k>=3, every set of k permutations on [n] hasa common subsequence of length at least n^{1/3}.  We show that, surprisingly, this lower bound is asymptotically optimal forall constant values of k. Specifically, we show that for any k>=3 and n>=k^2there exists a set of k permutations on [n] in which the longest commonsubsequence has length at most 32(kn)^{1/3}. The proof of the upper bound isconstructive, and uses elementary algebraic techniques.

Longest Common Subsequences in Sets of Permutations