Need help determining the largest intersection between two distinct Steiner Triple Systems of Order 7

Question

I have already found a solution where three of the blocks are the same, and I have proved that where five or six blocks are the same then the remaining blocks are forced, but I am having trouble showing the case of |$A\cap B$|= 4 either way.

I am guessing that $|A\cap B|$ can't be equal to 4, but am not sure where to start on my proof.

Any advice would be greatly appreciated.

Answer 1

Your guess is right. Up to isomorphism, there are only two possible configurations for the four blocks. If three blocks meet at a common point, the partial Steiner system is isomorphic to $\{123,\ 145,\ 167,\ 246\};$ otherwise it's isomorphic to $\{124,\ 136,\ 235,\ 456\}.$ In either case, the remaining three blocks are determined.