On the possible volume of $mu$-$(v,k,t)$ trades

‎A $mu$-way $(v,k,t)$ $trade$ of volume $m$ consists of $mu$‎ disjoint collections $T_1$‎, ‎$T_2‎, ‎dots T_{mu}$‎, ‎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 $T_i (1leq‎ ‎ileq mu)$‎. ‎In other words any pair of collections ${T_i,T_j}$‎, ‎$1leq i‎In 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 $m,mgeq2$‎. ‎(ii) 3-way $(v,k,2)$ trades of‎ ‎each volume $m,mgeq6$ 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 ‎$kgeq4$‎. ‎Also the spectrum of 3-way $(v,k,2)$ Steiner trades for‎ ‎$k=3$ and 4 are specified‎.