The fine triangle intersection problem for -designs

Let F i n ( v ) = { ( s , t ) : ∃  a pair of  ( K 4 − e ) -designs of order  v  intersecting in  s  blocks and  2 s + t  triangles } . Let A d m ( v ) = { ( s , t ) : s + t ≤ b v , s ∈ J ( v ) , 2 s + t ∈ J T ( v ) } ∖ { ( b v − 3 , 1 ) } , where J ( v ) (or J T ( v ) ) denotes the set of positive integers s (or t ) such that there exists a pair of ( K 4 − e ) -designs of order v intersecting in s blocks (or t triangles), and b v = v ( v − 1 ) / 10 . It is established that F i n ( v ) = A d m ( v ) for any integer v ≡ 0 , 1 ( mod 5 ) , v ≥ 6 and v ≠ 10 , 11 .