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In a finite projective plane, each two points appear together in exactly one line, and each two lines intersect in exactly one point. It is known that, if each line contains $n+1$ points, then the total number of points and of lines is $n^2+n+1$.

What is known about a generalization of this concept, in which each two points appear together in exactly $k$ lines, and each two lines intersect in exactly $k$ points? In particular, for what values of $n,k$ such planes exist, and what is the number of lines and points in that case?

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    $\begingroup$ A set where every two points appear together in exactly $k$ lines is an example of a design, sometimes called block design, or combinatorial design. I don't know about the converse condition, but I expect people have thought about it. Googling for designs would be a good place to start I guess $\endgroup$ – Vincent Jun 7 '17 at 9:44
  • $\begingroup$ This was just what I was looking for: en.wikipedia.org/wiki/Block_design. Thanks! $\endgroup$ – Erel Segal-Halevi Jun 7 '17 at 10:03
  • $\begingroup$ An even more general generalization is combinatorial design: en.wikipedia.org/wiki/Combinatorial_design $\endgroup$ – Erel Segal-Halevi Jun 7 '17 at 11:54
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    $\begingroup$ Given that you have conditions on both point pairs and line pairs, I think you should especially look at symmetric balanced incomplete block designs. The Bruck-Ryser-Chowla theorem gives necessary condtions for such things to exist. $\endgroup$ – Nick Gill Jun 7 '17 at 12:35
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Let me adjust notation slightly -- the $k$ in the original post is more usually a $\lambda$ in the literature. Thus the concept you want is this:

Definition. A symmetric $2-(v,k,\lambda)$ design is a pair $(\Omega, \mathcal{B})$ where $\Omega$ is a set of size $v$ and $\mathcal{B}$ is a set of $k$-subsets of $\Omega$ such that:

  • any 2 points of $\Omega$ lie in $\lambda$ elements of $\mathcal{B}$;
  • any 2 elements of $\mathcal{B}$ intersect in $\lambda$ elements of $\Omega$.

A simple counting argument asserts that an object has the property that $b=|\mathcal{B}|=v$. If you want to know when these things exist, then the following theorem should be your starting point:

The Bruck-Ryser-Chowla Theorem. If a symmetric $2-(v,k,\lambda)$ design exists, then

  • if $v$ is even, then $k-\lambda$ is a square;
  • if $v$ is odd, then the following Diophantine equation has a nontrivial solution: $$x^2-(k-\lambda)y^2 - (-1)^{(v-1)/2}\lambda z^2=0.$$

More is known in special cases. For instance there is a famous result of Lam, using a computer, that asserts that a symmetric $2-(111,11,1)$ design does not exist (there is no projective plane of order $10$).

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  • $\begingroup$ Thanks! Is any asymptotic existence result known? I mean a result of the form "for any $k,\lambda$, there exist infintely many $v$ such that a symmetric 2-$(v,k,\lambda)$ design exists"? $\endgroup$ – Erel Segal-Halevi Jun 7 '17 at 14:12
  • $\begingroup$ Unless I'm being a silly sausage, a counting argument tells you that $v=\frac{k^2-k}{\lambda}+1$ so the possibilities for $v$ are very restricted for any fixed $k$. In particular, there are finitely many... $\endgroup$ – Nick Gill Jun 7 '17 at 19:15

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