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 a 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 an arbitrary position in \(s\) 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 \(\operatorname{Sat}(u, n)\) denote the minimum possible length of a \(u\)-saturated sequence on an alphabet of size \(n\), and let the semisaturation function \(\operatorname{Ssat}(u, n)\) denote the minimum possible length of a \(u\)-semisaturated sequence on an alphabet of size \(n\). For alternating sequences, we determine both the saturation function and the semisaturation function up to a constant multiplicative factor. We show for every sequence that the semisaturation function is always either \(O(1)\) or \(\Theta(n)\). For the saturation function, we show that every sequence \(u\) has either \(\operatorname{Sat}(u, n) \ge n\) or \(\operatorname{Sat}(u, n) = O(1)\). For every sequence with \(2\) distinct letters, we show that the saturation function is always either \(O(1)\) or \(\Theta(n)\).