The triangle intersection problem for S(2,4,v)S(2,4,v) designs

In this paper the triangle intersection problem for S(2,4,v)S(2,4,v) designs is investigated. Let tv=v(v−1)/3tv=v(v−1)/3 and IT(v)={0,1,…,tv−30}∪{tv−27,tv−24,tv−18,tv}IT(v)={0,1,…,tv−30}∪{tv−27,tv−24,tv−18,tv}. Let JT(v)={s|JT(v)={s| there exist two S(2,4,v)S(2,4,v) designs with ss common triangles}}. We show that for any positive integer v≡1,4(mod12), JT(v)=IT(v)JT(v)=IT(v) when v≥121v≥121, and IT(v)∖{tv−33}⊆JT(v)⊆IT(v)IT(v)∖{tv−33}⊆JT(v)⊆IT(v) when 49≤v≤11249≤v≤112.