Sequence saturation

In this paper, we introduce saturation and semisaturation functions of sequences, and we prove a number of fundamental results about these functions. Given any forbidden sequence u with r distinct letters, we say that a sequence s on a given alphabet is u-saturated if s is r-sparse, u-free, and adding any letter from the alphabet to s violates r-sparsity or induces a copy of u. We say that s is u-semisaturated if s is r-sparse and adding any letter from the alphabet to s violates r-sparsity or induces a new copy of u. Let the saturation function Sat(u, n) denote the minimum possible length of a u-saturated sequence on an alphabet of size n, and let the semisaturation function Ssat(u, n) denote the minimum possible length of a u-semisaturated sequence on an alphabet of size n. For alternating sequences of the form a b a b …, we determine the saturation functions up to a multiplicative factor of 2, and we determine the semisaturation functions up to the leading term. We demonstrate a dichotomy for the semisaturation functions of sequences: for any sequence u, we have Ssat(u, n) = O(1) if and only if the first letter and the last letter of u each occur exactly once, and otherwise we have Ssat(u, n) = Θ(n). For the saturation function, we show that every sequence u has either Sat(u, n) ≥ n for every positive integer n or Sat(u, n) = O(1). We prove that every sequence u in which every letter occurs at least twice has Sat(u, n) ≥ n, and we show that Sat(u, n) = Θ(n) or Sat(u, n) = O(1) for every sequence u with 2 distinct letters.