论文标题
块设计的埃文斯风格的结果
An Evans-style result for block designs
论文作者
论文摘要
For positive integers $n$ and $k$ with $n \geq k$, an $(n,k,1)$-design is a pair $(V, \mathcal{B})$ where $V$ is a set of $n$ points and $\mathcal{B}$ is a collection of $k$-subsets of $V$ called blocks such that each pair of points occur together in exactly one block.如果我们削弱了这种条件以仅要求每对两点在最多一个块中一起出现，则结果对象是部分$（n，k，1）$ - 设计。部分$（n，k，1）$ - design $（v，\ nathcal {a}）$是a（完整的）$（n，k，1）$ - design $（v，\ nathcal {b}）$，以便$ \ mathcal {a} {a} \ subseteq \ subseteq \ mathcal {b} $。在这里，对于所有足够大的$ n $，我们准确确定了无法完全的部分$（n，k，1）$ - 设计中的最小块数。该结果让人联想到埃文斯（Evans）现已推荐的部分拉丁正方形的猜想。我们还证明了一些与$ k_k $的副本的边缘分解有关的一些相关结果。
For positive integers $n$ and $k$ with $n \geq k$, an $(n,k,1)$-design is a pair $(V, \mathcal{B})$ where $V$ is a set of $n$ points and $\mathcal{B}$ is a collection of $k$-subsets of $V$ called blocks such that each pair of points occur together in exactly one block. If we weaken this condition to demand only that each pair of points occur together in at most one block, then the resulting object is a partial $(n,k,1)$-design. A completion of a partial $(n,k,1)$-design $(V,\mathcal{A})$ is a (complete) $(n,k,1)$-design $(V,\mathcal{B})$ such that $\mathcal{A} \subseteq \mathcal{B}$. Here, for all sufficiently large $n$, we determine exactly the minimum number of blocks in an uncompletable partial $(n,k,1)$-design. This result is reminiscent of Evans' now-proved conjecture on completions of partial latin squares. We also prove some related results concerning edge decompositions of almost complete graphs into copies of $K_k$.
