I am now study some theorems of block design. I have a question about the derived designs.
Let $B$ be the oringinal design $t-(v,k, \lambda)$. Suppose we omit one of the points, say $P$, then we have two sets of blocks which remain: First, $B_{1}$, consisting of the $\lambda_{1,0}$ blocks of $k$ points that did not contain $P$. Second, $B_{2}$, consisting $\lambda_{1,1}$ blocks of $k-1$ points. These are called derived designs.
The theorem says that, $B_{1}$ form a $(t-1)-(v-1,k,\lambda_{t,t-1})$ design with block intersection $\lambda^{(1)}_{i,j}=\lambda_{i+1,j}$. $B_{2}$ form a $(t-1)-(v-1,k-1,\lambda)$ design with block intersection $\lambda^{(2)}_{i,j}=\lambda_{i+1,j+1}$.
Let us consider the $B=2-(7,3,1)$ design, then we have $B_{1}=1-(6,2,1)$ with $\lambda^{(1)}_{i,j}=\lambda_{i+1,j}$. If we consider $\lambda^{(1)}_{2,0}$, I think that $\lambda^{(1)}_{2,0}=1$, but the theorem says that $\lambda^{(1)}_{2,0}=\lambda_{3,0}=0$. What is the problem about this theorem? Or could someone give me the proof of this theorem? THANKS SO MUCH.