Consider a (simple undirected) graph $G$ with set of vertices $V=A\cup B$ with $|B|=30$.
(1) Every vertex in $A$ has an edge to exactly $3$ vertices in $B$.
(2) Every vertex in $B$ has an edge to exactly $4$ other vertices in $B$.
(3) For any two vertices in $B$ connected by an edge, some vertex in $A$ is connected to both of them.
What is the minimum size of $A$?
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The number of edges in $B$ is $2|B|=60$. The number of edges between $A$ and $B$ is $3|A|$. Condition (3) implies that for each of the $60$ edges in $B$, some vertex in $A$ is connected to both endpoints of the edge. Since every vertex in $A$ is connected to $3$ vertices in $B$ (hence also $3$ pairs of vertices), $|A|\geq 20$.
Is this the optimal bound, and if so, how can we construct an example?