The number of repeated blocks in twofold triple systems

In this paper, we give a complete answer to the following question: Given an integer υ ≡ 0 or 1 (mod 3) and an integer k, does there exist a twofold triple system of order υ with exactly k repeated triples? In particular, we prove the following theorem: If υ ≡ 0 or 4 (mod 6), υ > 12, then there exists a twofold triple system of order υ having exactly k repeated triples if and only if k ϵ I′υ, where I′υ = {0, 1,…, sυ − 2, sυ} if υ ≡ 0 (mod 4), I′υ = {0, 1,…, sυ − 1} if υ ≡ 2 (mod 4), and sυ = υ(υ − 4)6.