Author/Authors :
Rashidi، Saeedeh نويسنده Alzahra University , , Soltankhah، Nasrin نويسنده Alzahra University ,
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.
