Some results on the Ryser design conjecture

Abstract

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 $A$ 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 rational number $\dfrac{r_1-1}{r_2-1}$. Main results of the present article are the following. For every block $A$, $r_1 \geq |A| \geq r_2$ (this improves an earlier known inequality $|A| \geq r_2$). If there is no small block (respectively no large block) in $\mathcal{D}$, then $D\leq -1$ (respectively $D\geq 0$). With an extra assumption $e_2 > e_1$ an earlier known upper bound on $v$ is improved from a cubic to a quadratic in $\lambda$. It is also proved that if $v \leq \lambda^2+ \lambda + 1$ and if $\rho$ equals $\lambda$ or $\lambda - 1$, then $\mathcal{D}$ is of Type-1. Finally a Ryser design with $ 2^n + 1$ points is shown to be of Type-1.