Some results on \lambda-design conjecture

Let v and \lambda be integers with ۰<\lambda\lambda, and not all k_j are equal.The only known examples of \lambda-designs are so called of type-۱ 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-۱. 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-۱ under the condition that |\lambda(x,y)-\lambda(x,y')|< ۲ \left(\dfrac{r-r^*}{r+r^*-۲}\right). Under this condition, we prove that if \dfrac{r-۱}{r^*-۱} \le ۳, then \lambda-design D is of type-۱. Also we prove that D has exactly two distinct block sizes.

Keywords:

\lambda-designs , Ryser-designs , symmetric designs , \lambda-design conjecture , type-۱ \lambda-designs