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 \(|A| < 2\lambda \) (respectively \(|A| > 2\lambda \)) and average if \(|A|=2\lambda \). Let D denote the integer \(e_1 - r_2\) and let \(\rho > 1\) denote the ratio of \(r_1-1\) and \(r_2-1\). 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.

甲Ryser设计\（{\ mathcal {d}} \）上v点是集合v的点集的适当子集（称为块）与v点，使得每两个块彼此相交\（\拉姆达\ ）点（并且\（\ lambda <v \）是固定数），并且至少有两个块大小。如果\（{\ mathcal {D}} \）的每个点都具有相同的复制编号（或等效地，所有块具有相同的大小），则设计\（{\ mathcal {D}} \}被称为对称设计。），并且每两个块在\（\ lambda \）中相交点。Ryser设计的唯一已知构造是通过对称设计的块互补。这样的Ryser设计称为Type-1的Ryser设计。这是Ryser-Woodall猜想的基础：“每个Ryser设计都是Type-1”。这种长期存在的推测已被证明在许多情况下都是有效的。令\ {{\ mathcal {D}} \}表示v阶，索引\（\ lambda \）和复制编号\（r_1，r_2 \）的Ryser设计。令\（e_i \）表示复制数为\（r_i \）（其中\（i = 1，2 \））的\ {{\ mathcal {D}} \}的点数。调用的块甲的\（{\ mathcal {d}} \）如果\（| A | <2 \ lambda \）（分别为\（| A |> 2 \ lambda \）），则为小（分别较大）；如果\（| A | = 2 \ lambda \），则取平均值。令D表示整数\（e_1-r_2 \）并令\（\ rho> 1 \）表示\（r_1-1 \）与\（r_2-1 \）的比率。本文的主要结果如下：建立Ryser设计集上的等价关系。对Ryser-Woodall的块互补过程进行了一些观察。结果表明，Ryser设计具有两种块大小，其中一种是平均块大小，其类型为Type-1。还表明，在假定在与给定Ryser设计等效的任何Ryser设计中不存在大块和小块的情况下，给定的Ryser设计必须为Type-1。