Show that a strongly regular graph with $28$ verticies each with valency $9$ such that adjacent pairs are not joined and non-adjacent pairs share exactly $4$ vertices does not exist.
A really bad proof is the following: If such an object existed, some clever spark would have found it, and despite an extensive search I have not found it in the literature. I have a toe curling argument involving 4 subsets of a 9 set and then some, that doesn't even convince me!
Is there a reasonably simple counting arguement that shows the non-existence of this? Failing that a reference to a computer search that shows it?
As usual my thanks in advance to all who offer a response. HK