The intersection spectrum of Skolem sequences and its applications to [formula omitted]-fold cyclic triple systems

Abstract

A Skolem sequence of order n is a sequence Sn=(s1,s2,…,s2n) of 2n integers containing each of the integers 1,2,…,n exactly twice, such that two occurrences of the integer j∈{1,2,…,n} are separated by exactly j−1 integers. We prove that the necessary conditions are sufficient for existence of two Skolem sequences of order n with 0,1,2,…,n−3 and n pairs in the same positions. Further, we apply this result to the fine structure of cyclic two-, three-, and four-fold triple systems, and also to the fine structure of λ-fold directed triple systems and λ-fold Mendelsohn triple systems.