Intersection of blocks of the symmetric BIBD $PG(d,q)$

Question

The definition of a Balance Incomplete block design $(v,k,\lambda)$-BIBD can be found here. It is a well known fact (also see the link above) that every two blocks of a symmetric $(v,k,\lambda)$-BIBD have exactly $\lambda$ elements in common. I was wondering if the following is also true:

Guess: If three distinct blocks of a symmetric $(v,k,\lambda)$-BIBD have at least $2$ elements in common, then they have exactly $\lambda$ elements in common.

Does that hold?

Answer 1

It doesn't seem to be true. Consider the following symmetric $(31,15,7)$-BIBD whose incidence matrix is obtained from the 5th tensor power of the matrix $\bigl(\begin{smallmatrix}1 & 1 \\ 1 & -1\end{smallmatrix}\bigr)$ by deleting the first row and column, and replacing $-1$ by $0$ everywhere: $$\begin{pmatrix} 0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0\\ 1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0\\ 0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1\\ 1&\!\!1&\!\!1&\!\!0&\!\!0&\!\!0&\!\!0&\!\!1&\!\!1&\!\!1&\!\!1&\!\!0&\!\!0&\!\!0&\!\!0&\!\!1&\!\!1&\!\!1&\!\!1&\!\!0&\!\!0&\!\!0&\!\!0&\!\!1&\!\!1&\!\!1&\!\!1&\!\!0&\!\!0&\!\!0&\!\!0\\ 0&\!\!1&\!\!0&\!\!0&\!\!1&\!\!0&\!\!1&\!\!1&\!\!0&\!\!1&\!\!0&\!\!0&\!\!1&\!\!0&\!\!1&\!\!1&\!\!0&\!\!1&\!\!0&\!\!0&\!\!1&\!\!0&\!\!1&\!\!1&\!\!0&\!\!1&\!\!0&\!\!0&\!\!1&\!\!0&\!\!1\\ 1&\!\!0&\!\!0&\!\!0&\!\!0&\!\!1&\!\!1&\!\!1&\!\!1&\!\!0&\!\!0&\!\!0&\!\!0&\!\!1&\!\!1&\!\!1&\!\!1&\!\!0&\!\!0&\!\!0&\!\!0&\!\!1&\!\!1&\!\!1&\!\!1&\!\!0&\!\!0&\!\!0&\!\!0&\!\!1&\!\!1\\ 0&\!\!0&\!\!1&\!\!0&\!\!1&\!\!1&\!\!0&\!\!1&\!\!0&\!\!0&\!\!1&\!\!0&\!\!1&\!\!1&\!\!0&\!\!1&\!\!0&\!\!0&\!\!1&\!\!0&\!\!1&\!\!1&\!\!0&\!\!1&\!\!0&\!\!0&\!\!1&\!\!0&\!\!1&\!\!1&\!\!0\\ 1&\!\!1&\!\!1&\!\!1&\!\!1&\!\!1&\!\!1&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!1&\!\!1&\!\!1&\!\!1&\!\!1&\!\!1&\!\!1&\!\!1&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0\\ 0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1\\ 1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1\\ 0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0\\ 1&\!\!1&\!\!1&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!1&\!\!1&\!\!1&\!\!1&\!\!1&\!\!1&\!\!1&\!\!1&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!1&\!\!1&\!\!1&\!\!1\\ 0&\!\!1&\!\!0&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!1&\!\!0&\!\!1&\!\!0\\ 1&\!\!0&\!\!0&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!1&\!\!1&\!\!0&\!\!0\\ 0&\!\!0&\!\!1&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!1&\!\!0&\!\!0&\!\!1\\ 1&\!\!1&\!\!1&\!\!1&\!\!1&\!\!1&\!\!1&\!\!1&\!\!1&\!\!1&\!\!1&\!\!1&\!\!1&\!\!1&\!\!1&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0\\ 0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1\\ 1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1\\ 0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0\\ 1&\!\!1&\!\!1&\!\!0&\!\!0&\!\!0&\!\!0&\!\!1&\!\!1&\!\!1&\!\!1&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!1&\!\!1&\!\!1&\!\!1&\!\!0&\!\!0&\!\!0&\!\!0&\!\!1&\!\!1&\!\!1&\!\!1\\ 0&\!\!1&\!\!0&\!\!0&\!\!1&\!\!0&\!\!1&\!\!1&\!\!0&\!\!1&\!\!0&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!1&\!\!0&\!\!1&\!\!0&\!\!0&\!\!1&\!\!0&\!\!1&\!\!1&\!\!0&\!\!1&\!\!0\\ 1&\!\!0&\!\!0&\!\!0&\!\!0&\!\!1&\!\!1&\!\!1&\!\!1&\!\!0&\!\!0&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!1&\!\!1&\!\!0&\!\!0&\!\!0&\!\!0&\!\!1&\!\!1&\!\!1&\!\!1&\!\!0&\!\!0\\ 0&\!\!0&\!\!1&\!\!0&\!\!1&\!\!1&\!\!0&\!\!1&\!\!0&\!\!0&\!\!1&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!1&\!\!0&\!\!0&\!\!1&\!\!0&\!\!1&\!\!1&\!\!0&\!\!1&\!\!0&\!\!0&\!\!1\\ 1&\!\!1&\!\!1&\!\!1&\!\!1&\!\!1&\!\!1&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!1&\!\!1&\!\!1&\!\!1&\!\!1&\!\!1&\!\!1&\!\!1\\ 0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0\\ 1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0\\ 0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1\\ 1&\!\!1&\!\!1&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!0&\!\!1&\!\!1&\!\!1&\!\!1&\!\!0&\!\!0&\!\!0&\!\!0&\!\!1&\!\!1&\!\!1&\!\!1&\!\!1&\!\!1&\!\!1&\!\!1&\!\!0&\!\!0&\!\!0&\!\!0\\ 0&\!\!1&\!\!0&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!1&\!\!0&\!\!1&\!\!0&\!\!0&\!\!1&\!\!0&\!\!1&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!1&\!\!0&\!\!0&\!\!1&\!\!0&\!\!1\\ 1&\!\!0&\!\!0&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!1&\!\!1&\!\!0&\!\!0&\!\!0&\!\!0&\!\!1&\!\!1&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!0&\!\!0&\!\!1&\!\!1\\ 0&\!\!0&\!\!1&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!1&\!\!0&\!\!0&\!\!1&\!\!0&\!\!1&\!\!1&\!\!0&\!\!1&\!\!0&\!\!0&\!\!1&\!\!1&\!\!0&\!\!0&\!\!1&\!\!0&\!\!1&\!\!1&\!\!0\end{pmatrix}$$ Observe that e.g. the 4th, 5th and 6th blocks have only 3 points in common (actually, almost any combination contradicts your claim)