On the possible volume of three way trades

A 3-way (v, k, t) trade of volume s consists of 3 disjoint collections T 1 , T 2 and T 3 , 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 T i ( 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.