On the existence of an S_lambda(2,4,v) which embeds an S(2,3,v-w) of maximum order for lambda\ge 2.

Let $H$ be a subgraph of $G$. An $H$-design $(X,{\Cal C})$ of order $v- w$, $0\le w\le v$, and index $\mu$ is embedded into a $G$-design $(V,{\Cal B})$ of order $v$ and index $\lambda$, if $\mu\le\lambda$, $X\subseteq V$ and there is an injective mapping $f:{\Cal C}\to{\Cal B}$ such that $B$ is subgraph of $f(B)$ for every $B\in{\Cal C}$. For every pair of positive integers $v$, $\lambda$, we determine the minimum value of $w$ such that there exists a balanced incomplete block design of order $v$, index $\lambda\ge 2$ and block-size 4 which embeds a $K_3$-design of order $v- w$, $0\le w\le v$, and index $\mu= 1$.