Some results on $\lambda$-design conjecture

Abstract

Let $v$ and $\lambda$ be integers with $0\lambda$, and not all $k_j$ are equal.The only known examples of $\lambda$-designs are so called of type-1 designs, which are obtained from symmetric designs by a certain complementation procedure. Ryser and Woodall had independently conjectured that all $\lambda$-designs are of type-1. Suppose $r$ and $r^*(r>r^*)$ are replication numbers of $D$ and for distinct points $x$ and $y$ of $D$, let $\lambda(x,y)$ denote the number of blocks of $X$ containing $x$ and $y$. In this paper we investigate the possibilities of $\lambda$-designs to be of type-1 under the condition that $|\lambda(x,y)-\lambda(x,y')|< 2 \left(\dfrac{r-r^*}{r+r^*-2}\right)$. Under this condition, we prove that if $ \dfrac{r-1}{r^*-1} \le 3$, then $\lambda$-design $D$ is of type-1. Also we prove that $D$ has exactly two distinct block sizes.

Keywords

• $\lambda$-designs
• ryser-designs
• symmetric designs
• $\lambda$-design conjecture
• type-1 $\lambda$-designs