Since I am unable to upload images (don't know why), assume that the circle is a rectangle
So, from condition (III), we get (Let % = Other People and # = Blank Space)
P W T
% # %
% # %
% % %
As P and T are adjacent to W
Then, from condition (V),
P W T
% # %
% # %
% R %
As there are supposed to be 4 persons between R and W
After that, from condition (VI), we see that there are 2 possibilities :
No. 1
P W T
V # %
% # %
% R %
OR
No. 2
P W T
% # V
% # %
% R %
Now, with condition (IV), possibility No. 2 can be eliminated (Big thanks to @Alconja for telling me)
As S is supposed to be the right of V which is not possible in No. 2. Hence, we continue with No.1
P W T
V # Q
S # %
% R %
Then, with condition (I) and (II), we see that there are 2 possibilities again :
No.1
P W T
V # Q
S # X
U R Y
OR
No. 2
P W T
V # Q
S # U
X R Y
Since, Y is not adjacent to X,
No. 2 is the only possibility
So, the final answer is
P W T
V # Q
S # U
X R Y