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.
Rashidi, S., & Soltankhah, N. (2014). On the possible volume of $\mu$-$(v,k,t)$ trades. Bulletin of the Iranian Mathematical Society, 40(6), 1387-1401.
MLA
S. Rashidi; N. Soltankhah. "On the possible volume of $\mu$-$(v,k,t)$ trades". Bulletin of the Iranian Mathematical Society, 40, 6, 2014, 1387-1401.
HARVARD
Rashidi, S., Soltankhah, N. (2014). 'On the possible volume of $\mu$-$(v,k,t)$ trades', Bulletin of the Iranian Mathematical Society, 40(6), pp. 1387-1401.
VANCOUVER
Rashidi, S., Soltankhah, N. On the possible volume of $\mu$-$(v,k,t)$ trades. Bulletin of the Iranian Mathematical Society, 2014; 40(6): 1387-1401.
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