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

Document Type : Research Paper


1 Alzahra Uni

2 Alzahra Uni.


‎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.


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