4 added 4 characters in body

A while ago I asked how to construct an infinite family of $(v,b,r,k,\lambda)$-designs satisfying $r=\lambda^{2}$ and got very good answers from Yuichiro Fujiwara and Ken W. Smith.

Now I'd like to up the ante and to generalize the question to $(r,\lambda)$-designs. Formally, we are talking here about a family $D$ of subsets of $\{1,2,\ldots,v\}$ such that:

(a) Each $i \in \{1,2,\ldots,v\}$ belongs to $r$ sets in $D$.

(b) Every two distinct $i,j \in \{1,2,\ldots,v\}$ belong together to $\lambda$ sets in $D$.

I am trying to find examples with $r=\lambda$, r=\lambda^{2}$, which is more difficult than in the case of$(v,k,\lambda)$designs because this is more off the beaten path, so even stand-alone examples will be greatly appreciated. P.S. As in the previous question, I require at least one pair of disjoint blocks, so most constructions based on symmetric designs will not apply here. 3 added 34 characters in body A while ago I asked how to construct an infinite family of$(v,b,r,k,\lambda)$-designs satisfying$r=\lambda^{2}$and got very good answers from Yuichiro Fujiwara and Ken W. Smith. Now I'd like to up the ante and to generalize the question to$(r,\lambda)$-designs. Formally, we are talking here about a family$D$of subsets of$\{1,2,\ldots,v\}$such that: (a) Each$i \in \{1,2,\ldots,v\}$belongs to$r$sets in$D$. (b) Every two distinct$i,j \in \{1,2,\ldots,v\}$belong together to$\lambda$sets in$D$. I am trying to find examples with$r=\lambda$, which is more difficult than in the case of$(v,k,\lambda)$designs because this is more off the beaten path, so even stand-alone examples will be greatly appreciated. P.S. As in the previous question, I require at least one pair of disjoint blocks, so most constructions based on symmetric designs will not apply here. 2 added 6 characters in body A while ago I asked how to construct an infinite family of$(v,b,r,k,\lambda)$-designs satisfying$r=\lambda^{2}$and got very good answers from Yuichiro Fujiwara and Ken W. Smith. Now I'd like to up the ante and to generalize the question to$(r,\lambda)$-designs. Formally, we are talking here about a family$D$of subsets of$\{1,2,\ldots,v\}$such that: (a) Each$i \in \{1,2,\ldots,v\}$belongs to$r$sets in$D$. (b) Every two distinct$i,j \in \{1,2,\ldots,v\}$belong together to$\lambda$sets in$D$. I am trying to find examples with$r=\lambda$, which is more difficult than in the case of$(v,k,\lambda)\$ designs because this is more off the beaten path, so even stand-alone examples will be greatly appreciated.

P.S. As in the previous question, I require at least one pair of disjoint blocks, so most constructions based in on symmetric designs will not workapply here.

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