@article { author = {Rashidi, S. and ‎Soltankhah, N.}, title = {On the possible volume of $\mu$-$(v,k,t)$ trades}, journal = {Bulletin of the Iranian Mathematical Society}, volume = {40}, number = {6}, pages = {1387-1401}, year = {2014}, publisher = {Iranian Mathematical Society (IMS)}, issn = {1017-060X}, eissn = {1735-8515}, doi = {}, abstract = {‎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 $T_i (1\leq‎ ‎i \leq \mu)$‎. ‎In other words any pair of collections $\{T_i,T_j\}$‎, ‎$1\leq i< j \leq \mu‎$ 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 $m,m\geq2$‎. ‎(ii) 3-way $(v,k,2)$ trades of‎ ‎each volume $m,m\geq6$ 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 ‎$k\geq4$‎. ‎Also the spectrum of 3-way $(v,k,2)$ Steiner trades for‎ ‎$k=3$ and 4 are specified‎.}, keywords = {$mu$-way $(v,k,t)$ trade‎,‎3-way $(v,2)$ trade‎,‎one-solely‎}, url = {http://bims.iranjournals.ir/article_571.html}, eprint = {http://bims.iranjournals.ir/article_571_a228d214fe2139e3f118bdf489628d23.pdf} }