The intersection spectrum of Skolem sequences and its applications to lambda fold cyclic triple systems, together with the Supplement

TL;DR

This paper investigates the intersection spectrum of Skolem sequences, establishes existence conditions for certain pairs, and applies these findings to analyze the structure of various cyclic and lambda-fold triple systems.

Contribution

It proves sufficiency of conditions for the existence of specific Skolem sequence pairs and applies these results to the structural analysis of multiple types of triple systems.

Findings

Established existence conditions for pairs of Skolem sequences with specific overlaps.

Applied Skolem sequence intersection results to analyze cyclic and lambda-fold triple systems.

Enhanced understanding of the structure of various triple systems through these applications.

Abstract

A Skolem sequence of order n is a sequence S_n=(s_{1},s_{2},...,s_{2n}) of 2n integers containing each of the integers 1,2,...,n exactly twice, such that two occurrences of the integer j in {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 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 lambda-fold directed triple systems and lambda-fold Mendelsohn triple systems. For a better understanding of the paper we added more details into a "Supplement".