As each person in the party is acquainted to three others, the number of acquaintance relations must be $\frac{1}{2} \cdot 8 \cdot 3 = 12$. Assume $A$ is one of the persons in the party. Denote by $B$, $C$ and $D$ the three persons $A$ knows. To meet condition (i), none of $B$, $C$ and $D$ know each other. So each of them has to know two persons in the set $S = \{E, F, G, H\}$. Up to now, we have used 9 acquaintances, so there are exactly 3 acquaintance relations between members $S$. Again, no three persons in $S$ are to know each other. On the other hand, if one of the persons, say $E$, is acquainted to all three others, $F$, $G$ and $H$, the set $\{A, F, G, H\}$ would violate condition (ii). So the only possibility is that the acquaintance relations are arranged in a linear manner: we may name the persons in such a manner that $E$ knows $F$, $F$ knows $G$ and $G$ knows $H$. Now $E$ knows exactly two persons in the set $T = \{B, C, D\}$. We may assume that they are $B$ and $C$. $F$ knows exactly one person in the set $T$ and this person must be $D$. $G$ cannot be acquainted with $D$, so his acquaintance in $T$ is either $B$ or $C$. If it is $B$, then $C$ and $D$ can be the acquaintances of $H$, and if it is $C$, then $H$ can be acquainted to $B$ and $D$. In fact, we can always name the members in $T$ to conform with the former alternative.

The construction has provided an arrangement consistent with (i); to see that (ii) is fulfilled, we just need to check for each member that among the four members not acquainted to the first one, there is at least one pair of acquaintances. Indeed: for $A$ such a pair is ($E$, $F$), for $B$, ($C$, $H$), for $C$, ($D$, $F$), for $D$, ($C$, $E$), for $E$, ($A$, $D$), for $F$, ($A$, $C$), for $G$, ($A$, $D$) and for $H$, ($A$, $B$). – We note that there is essentially one solution, up to a renaming of the members.