Henry is right. The -ki word is an adjective attribute. That is why I also find

                         Hikmet´in dört yaşında oğlu var

strange. We have obviously missed something here. (That is probably why I often think Turks tend to forget the -ki´s...)

I find it difficult to see dört yaşında oğlu as the object here. If that is the object where is the subject then? Could it be that it has to do with the indefiniteness of that constituent (var/yok sentences typically introduce items for the first time)?

What if there was a real object:

                          ?Hikmet dört yaşında oğlunu bize tanıttı     ?

Dear tunci I think subject here is : Hikmet´in dört yaşında oğlu. We cant seperate it. I am thinking about why sometimes we use -ki and why sometimes we dont use it. When I have an answer I will post it

Dearest Gokuyum, I considered it  as  "to have "  [sahip olmak], therefore to me "Hikmet" is the subject who is having 4 year old son.   I mean, Hikmet is the doer who is performing the action of  "having 4 year old son " 

However, If we think your way, it makes sense too. Since " there is  a son of Hikmet who is 4 year old son " , In other words, " 4 year old son of Hikmet exists" , so the whole lot except "var [exists] " could  be considered as a subject. Your thinking is right.

 Thank you for pointing out this , Üstat.

 Hikmet dört yaşındaki oğlunu bize tanıttı     ?

We have to add  -ki  to describe his son [his son´s age] . Hikmet introduced his son whois 4 year old.

Otherwise the sentence may still make sense but semantically impossible sentence. Since it may sound like " Hikmet introduced us his son while he [Hikmet] was 4 year old. "  Logically it is impossible that Hikmet can have a child when he was 4 year old. Since sometimes we Turks drop the - iken , such as ;

Hikmet dört yaşında[y]ken ] piyano çalmaya başladı.

Hikmet started to play piano when he was 4 year old.