The triangle intersection problem for designs

In this paper the triangle intersection problem for S ( 2 , 4 , v ) designs is investigated. Let t v = v ( v − 1 ) / 3 and I T ( v ) = { 0 , 1 , … , t v − 30 } ∪ { t v − 27 , t v − 24 , t v − 18 , t v } . Let J T ( v ) = { s | there exist two S ( 2 , 4 , v ) designs with s common triangles } . We show that for any positive integer v ≡ 1 , 4 ( mod 12 ) , J T ( v ) = I T ( v ) when v ≥ 121 , and I T ( v ) ∖ { t v − 33 } ⊆ J T ( v ) ⊆ I T ( v ) when 49 ≤ v ≤ 112 .