The 3-way intersection problem for S(2, 4, v) designs

In this paper the 3-way intersection problem for $S(2,4,v)$ designs is investigated. Let $b_v=\frac{v(v-1)}{12}$ and $I_3[v]=\{0,1,\mathrm{.}\mathrm{.}\mathrm{.},b_v\}\setminus \{b_v-7,b_v-6,b_v-5,b_v-4,b_v-3,b_v-2,b_v-1\}$. Let $J_3[v]=\{k\mathrm{\mid }$ there exist three $S(2,4,v)$ designs with $k$ same common blocks$\}$. We show that $J_3[v]\subseteq I_3[v]$ for any positive integer $v\equiv 1,4(\mathrm{m}\mathrm{o}\mathrm{d}12)$ and $J_3[v]=I_3[v]$, for $v\ge 49$ and $v=13$. We find $J_3[16]$ completely. Also we determine some values of $J_3[v]$ for $v=25,28,37$ and 40.