ON THE POSSIBLE VOLUME OF μ-(v,k,t) TRADES.

A μ-way (v, k, t) trade of volume m consists of μ disjoint collections T1, T2, . . . Tμ, each of m 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 ≤ μ). In other words any pair of collections {Ti, Tj}, 1 ≤ i < j ≤ μ is a (v, k, t) trade of volume m. In this paper we investigate the existence of ≥-way (v, k, t) trades and prove the existence of. (i) 3-way (v, k, 1) trades Steiner trades) of each volume m,m ≥ 2. (ii) 3-way (v, k, 2) trades of each volume m,m ≥ 6 except possibly m = 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. [ABSTRACT FROM AUTHOR]