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

A mu-way (v, k, t) trade of volume m consists of mu disjoint collections T-1, T-2, . . .T-mu, 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 T-i (1 <= i <= mu). In other words any pair of collections {T-i, T-j}, 1 <= i < j <= mu is a (v, k, t) trade of volume m. In this paper we investigate the existence of mu-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.