  An avoidance problem of configurations in 4-cycle systems is investigated by
generalizing the notion of sparseness, which is originally from Erd\H{o}s'
r-sparse conjecture on Steiner triple systems. A 4-cycle system of order v,
4CS(v), is said to be r-sparse if for every integer j satisfying 1 < j < r+1 it
contains no configurations consisting of j 4-cycles whose union contains
precisely j+3 vertices. If an r-sparse 4CS(v) is also free from copies of a
configuration on two 4-cycles sharing a diagonal, called the double-diamond, we
say it is strictly r-sparse. In this paper, we show that for every admissible
order v there exists a strictly 4-sparse 4CS(v). We also prove that for any
positive integer r > 1 and sufficiently large integer v there exists a constant
number c such that there exists a strictly r-sparse 4-cycle packing of order v
with cv^2 4-cycles.
