The intersection spectrum of Skolem sequences and its applications to -fold cyclic triple systems

A Skolem sequence of order n is a sequence S n = ( s 1 , s 2 , … , s 2 n ) of 2 n 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.