Suppose the universe is infinitely big and you decided to travel throughout a line you randomly choosed. Does that mean you will travel forever without coming the same place you have been in the past?
It would be difficult to say, as it depends on your definition of "the same place".
If by "place" you mean space-time points then yes, eventually you would come back to where you started since there are an infinite number of such places in an infinitely large universe. However if you consider time instead (e.g., because you want to observe changes over time) this may not necessarily hold true for all times.
In particular, suppose you start at t=0 from some position A with respect to which you can define future events F(t')>F(t), i.e., something happens before something else. Then after traveling along the random line you chose, you reach time T', when things happen again like they did at time 0 or earlier. But now starting at location B, one could imagine that we arrive first somewhere between positions A and B later than someone who followed the path backwards but still arrived first because he/she was already present when what happened next happened according to our definition. So depending on how you choose these locations, you could end up having traveled both ways around the circle going through exactly the same set of points several times - yet different people might experience them differently due to their initial conditions. This doesn't contradict me saying that we don't know whether we'll ever come back to where we were previously though!