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

A $μ$-way $(v,k,t)$ $trade$ of volume $m$ consists of $μ$ disjoint collections $T_1$, $T_2, \dots T_μ$, 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 $T_i\ (1\leq i\leq μ)$. In other words any pair of collections $\{T_i,T_j\}$, $1\leq i<j \leq μ$ is a $(v,k,t)$ trade of volume $m$. In this paper we investigate the existence of $μ$-way $(v,k,t)$ trades and also we 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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