Skolem and Rosa rectangles and related designs

A Skolem sequence of order n is a sequence S n = ( s 1 , s 2 , … , s 2 n ) of 2 n integers such that (1) for all k ∈ { 1 , 2 , … , n } there are exactly two terms s i , s j such that s i = s j = k , and (2) if s i = s j = k and i < j , then j − i = k . olem sequences S n , S n ′ are disjoint if s i = s j = k = s t ′ = s u ′ implies that { i , j } ≠ { t , u } , for all k = 1 , 2 , … , n . For example, the two Skolem sequences of order four 1 , 1 , 4 , 2 , 3 , 2 , 4 , 3 and 2 , 3 , 2 , 4 , 3 , 1 , 1 , 4 are disjoint. A set of m pairwise disjoint Skolem sequences forms a Skolem rectangle of strength m . The above sequences then form a Skolem rectangle of strength two: , 4 , 2 , 3 , 2 , 4 , 3 , 2 , 4 , 3 , 1 , 1 , 4 . roduce several new constructions for Skolem and Rosa rectangles then we apply them to generate simple cyclic triple systems and disjoint cyclic triple systems. For example, for certain constants C 1 , C 2 we obtain ⌊ log 3 ( 2 n + C 1 ) ⌋ + C 2 disjoint Skolem sequences of order n . Finally, we obtain the analogous results for hooked Skolem and Rosa sequences.