EDIT: I submitted a less than helpful response the first time. Here is my proof in this edit.
When all $17$ people within a group know $4$ people from the group, then there are $34$ friend pairings.
In the above diagram, ensuring that everyone is at least a friend of a friend requires $52$ pairings "so far" just for persons $1$ through $5$. I have only worked the requirement for persons $1$ to $5$ because it already exceeds the requirement of knowing exactly $4$ others.
Every person in the group doesn’t personally know $12$ others in the group. But for there to be a possibility of sharing a friend with all $12$ others, every person’s $4$ friends must between them know all the other $12$.
In the diagram above, person $1$ personally knows $4$ others $(2,3,4,5)$. And between this $4$, they know all the other $12$ people ($6$ through $17$). But the same situation must exist for the friends of $1$, ($2,3,4$ and $5$). So, on the chart this requirement has been filled in where each set of $4$ friends for $2,3,4,5$ must know their corresponding other $12$. When this is done however, the number of friends for some of the people exceeds $4$. Not only that, but ensuring everyone is a least a friend of a friend hasn't been done for all $17$ in the group.
These are the $5$ acquaintance pairings so far:
$17 (8,11,12,13,14); 12 (6,7,8,10,17); 7(9,12,13,14,15)$
$16 (8,9,10,11,14); 11 (6,13,15,16,17); 6 (9,10,11,12,15)$
$15 (6,7,8,11,14); 10 (6,12,13,14,16)$
$14 (7,10,15,16,17); 9 (6,7,8,13,16)$
$13 (7,9,10,11,17); 8 (9,12,15,16,17)$
Therefore, for all unacquainted people to share a common friend, the unacquainted people have to know more than $4$ people. Hence with each person only knowing $4$ others, there will always be at least two people who don’t know each other and do not share a common friend.
A follow up question could be, what is the least number of acquaintances each person must have to ensure that everyone is at least a friend of a friend?