A Ryser design $\mathcal{D}$ on $v$ points is a collection of $v$ proper subsets (called blocks) of a point-set with $v$ points such that every two blocks intersect each other in $\lambda$ points (and $\lambda < v$ is a fixed number) and there are at least two block sizes. A design $\mathcal{D}$ is called a symmetric design, if every point of $\mathcal{D}$ has the same replication number (or equivalently, all the blocks have the same size) and every two blocks intersect each other in $\lambda$ points. The only known construction of a Ryser design is via block complementation of a symmetric design. Such a Ryser design is called a Ryser design of Type-1. This is the ground for the Ryser-Woodall conjecture: "every Ryser design is of Type-1". This long standing conjecture has been shown to be valid in many situations. Let $\mathcal{D}$ denote a Ryser design of order $v$, index $\lambda$ and replication numbers $r_1,r_2$. Let $e_i$ denote the number of points of $\mathcal{D}$ with replication number $r_i$ (with $i = 1, 2$). Call a block $A$ of $\mathcal{D}$ small (respectively large) if $|A| 2\lambda$) and average if $|A|=2\lambda$. Let $D$ denote the integer $e_1 - r_2$ and let $\rho> 1$ denote the rational number $\dfrac{r_1-1}{r_2-1}$. Main results of the present article are the following: An equivalence relation on the set of Ryser designs is established. Some observations on the block complementation procedure of Ryser-Woodall are made. It is shown that a Ryser design with two block sizes one of which is an average block size is of Type-1. It is also shown that, under the assumption that large and small blocks do not coexist in any Ryser design equivalent to a given Ryser design, the given Ryser design must be of Type-1.

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关于 Ryser 设计猜想的一些结果

$v$ 点上的 Ryser 设计 $\mathcal{D}$ 是具有 $v$ 点的点集的 $v$ 真子集（称为块）的集合，使得每两个块在 $\lambda 中彼此相交$ 点（并且 $\lambda < v$ 是一个固定数字）并且至少有两个块大小。一个设计 $\mathcal{D}$ 被称为对称设计，如果 $\mathcal{D}$ 的每个点都具有相同的复制数（或等效地，所有块具有相同的大小）并且每两个块彼此相交在 $\lambda$ 点。Ryser 设计的唯一已知结构是通过对称设计的块互补。这种 Ryser 设计称为 Type-1 的 Ryser 设计。这就是 Ryser-Woodall 猜想的基础：“每个 Ryser 设计都是 Type-1”。这个长期存在的猜想已被证明在许多情况下都是有效的。让 $\mathcal{D}$ 表示阶 $v$、索引 $\lambda$ 和复制数 $r_1,r_2$ 的 Ryser 设计。让 $e_i$ 表示 $\mathcal{D}$ 的点数，复制数为 $r_i$（$i = 1, 2$）。如果 $|A|，则调用 $\mathcal{D}$ 的块 $A$ 小（分别大）2\lambda$) 和平均值，如果 $|A|=2\lambda$。让 $D$ 表示整数 $e_1 - r_2$，让 $\rho> 1$ 表示有理数 $\dfrac{r_1-1}{r_2-1}$。本文的主要结果如下： 建立了 Ryser 设计集的等价关系。对 Ryser-Woodall 的块互补过程进行了一些观察。结果表明，具有两种块大小（其中一种是平均块大小）的 Ryser 设计属于类型 1。还表明，