Some results on the Ryser design conjecture

A Ryser design has equally many points as blocks with the provision that every two blocks intersect in a fixed number of points λ . An improper Ryser design has only one replication number and is thus symmetric design. A proper Ryser design has two replication numbers. The only known construction of a Ryser design is the complementation of a symmetric design. Such a Ryser design is called a Ryser design of type 1. Let D denote a Ryser design of order v , index λ and replication numbers r 1 , r 2 . Let e i denote the number of points of D with replication number r i (with i = 1 , 2 ). Call a block A small (respectively large) if| A | < 2 λ (respectively| A | > 2 λ ) and average if| A | = 2 λ . Let D denote the integer e 1 − r 2 and let ρ > 1 denote the rational numberr 1 − 1r 2 − 1 . Main results of the present article are the following. For every block A ,r 1 ≥ | A | ≥ r 2(this improves an earlier known inequality| A | ≥ r 2 ). If there is no small block (respectively no large block) in D , then D ≤ − 1 (respectively D ≥ 0 ). With an extra assumption e 2 > e 1 an earlier known upper bound on v is improved from a cubic to a quadratic in λ . It is also proved that if v ≤ λ 2 + λ + 1 and if ρ equals λ or λ − 1 , then D is of type 1. Finally, a Ryser design with 2 n + 1 points is shown to be of type 1.