Some results on $\lambda$-design conjecture

Let $v$ and $\lambda$ be integers with $0<\lambda<v$. A $\lambda$-design $D$ is a pair $(X, \mathcal{A})$, where $X$ is a finite set with $v$ elements called points and $\mathcal{A}$ is a family of subsets of $X$ called blocks, with $|\mathcal{A}|=|X|$ such that (1) for all $B_i, B_j\in \mathcal{A},$ $i\neq j,$ $|B_i\cap B_j|=\lambda$; (2) for all $B_j\in \mathcal{A},$ $|B_j|=k_j>\lambda$, 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.