On the possible volume of three way trades

A 3-way (v, k, t) trade of volume s consists of 3 disjoint collections T1,T2 and T3, each of s blocks, such that for every t-subset of v-set V, the number of blocks containing this t-subset is the same in each Ti (1⩽i⩽3). In this paper we prove the existence of: (i) 3-way (v, k, 1) trades (Steiner trades) of each volume s,s⩾2. (ii) 3-way (v, k, 2) trades of each volume s,s⩾6 except s=7. We establish the non-existence of 3-way (v, 3, 2) trade of volume 7. It is shown that the volume of a 3-way (v, k, 2) Steiner trade is at least 2k for k⩾4. Also the spectrum of 3-way (v,k,2) Steiner trades for k=3 and 4 are specified.