3路$(v,k,2)$ Steiner交易谱的完善
A Completion of the spectrum of 3-way $(v,k,2)$ Steiner trades
摘要：
卷$m$的3路$(v,k,t)$交易$T$由三个成对不相交的集合$T_1$、$T_2$和$T_3$组成，每个$m$块的大小为$k$，使得对于$v$集合$V$的每个$t$ -子集，包含该$t$ -子集的块的数量在$1\leq i\leq 3$的每个$T_i$中相同。如果在$1\leq i\leq 3$的每个$T_i$中，发现的任何$t$ -子集（$T$）最多出现一次，则$T$称为3路$(v,k,t)$ Steiner交易。我们试图通过应用一些块设计，如BIBD、RB、GDD、RGDD和$r\times s$填充网格块，来完成3路$(v,k,2)$ Steiner交易的频谱$S_{3s}(v,k)$，即所有可能的交易量大小的集合。以前，我们得到了一些结果，存在一些3路$(v,k,2)$ Steiner交易。特别是，我们证明了当$12(k-1)\leq m$对$15\leq k$时，存在$m$交易量的3向$(v,k,2)$ Steiner交易（Rashidi和Soltankhah，2016）。现在，我们证明了这个结论对于$k\leq 14$也是正确的。
英文摘要：
A 3-way $(v,k,t)$ trade $T$ of volume $m$ consists of three pairwise disjoint collections $T_1$, $T_2$ and $T_3$, 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 $1\leq i\leq 3$. If any $t$-subset of found($T$) occurs at most once in each $T_i$ for $1\leq i\leq 3$, then $T$ is called 3-way $(v,k,t)$ Steiner trade. We attempt to complete the spectrum $S_{3s}(v,k)$, the set of all possible volume sizes, for 3-way $(v,k,2)$ Steiner trades, by applying some block designs, such as BIBDs, RBs, GDDs, RGDDs, and $r\times s$ packing grid blocks. Previously, we obtained some results about the existence some 3-way $(v,k,2)$ Steiner trades. In particular, we proved that there exists a 3-way $(v,k,2)$ Steiner trade of volume $m$ when $12(k-1)\leq m$ for $15\leq k$ (Rashidi and Soltankhah, 2016). Now, we show that the claim is correct also for $k\leq 14$.
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