关于$d$ -齐次$3$ -路Steiner交换的存在性
On the existence of $d$-homogeneous $3$-way Steiner trades
摘要：
卷$m$的$μ$ -路$(v, k, t)$交易$T = \{T_{1} , T_{2}, . . ., T_μ \}$由$μ$不相交集合$T_{1}, T_{2}, \ldots, T_μ$组成，每个$m$块的大小为$k$，使得对于$v$ -集$V$的每个$t$ -子集，包含该$t$ -子集的块的数目在每个$T_{i}$中相同（对于$1 \leq i \leq μ$）.如果发现的$(T)$的任何$t$子集在$T_{1}$ $(T_{j},\ j \geq 2)$中最多出现一次，则$μ$-路$(v, k, t)$交易称为$μ$ -路$(v, k, t)$ Steiner交易.如果$V$的每个元素都精确地出现在$T_{1}$ $(T_{j},~ j \geq 2)$的$d$块中，则$μ$方式的$(v,k,t)$交易称为$d$ -同构.在本文中，我们描述了$3$ -方式$3$ -均匀$(v,3,2)$施泰纳交易量$v$。此外，我们还展示了如何构建一个$3$ -路$d$ -均匀$(v,3,2)$施泰纳贸易为$d\in \{4,5,6\}$，除了七个小值的$v$。
英文摘要：
A $μ$-way $(v, k, t)$ trade $T = \{T_{1} , T_{2}, . . ., T_μ \}$ of volume $m$ consists of $μ$ disjoint collections $T_{1}, T_{2}, \ldots, T_μ$, each of $m$ blocks of size $k$, such that for every $t$-subset of $v$-set $V$ the number of blocks containing this $t$-subset is the same in each $T_{i}$ (for $1 \leq i \leq μ$). A $μ$-way $(v, k, t)$ trade is called $μ$-way $(v, k, t)$ Steiner trade if any $t$-subset of found$(T)$ occurs at most once in $T_{1}$ $(T_{j},\ j \geq 2)$. A $μ$-way $(v,k,t)$ trade is called $d$-homogeneous if each element of $V$ occurs in precisely $d$ blocks of $T_{1}$ $(T_{j},~ j \geq 2)$. In this paper we characterize the $3$-way $3$-homogeneous $(v,3,2)$ Steiner trades of volume $v$. Also we show how to construct a $3$-way $d$-homogeneous $(v,3,2)$ Steiner trade for $d\in \{4,5,6\}$, except for seven small values of $v$.
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