Auto Insurance Claims: Accident with no tickets involved, claims suit, oncoming car

Question
I was in an accident recently where I was going straight through a stoplight and an oncoming car turned left directly in front of me resulting in a collision.  I had a green light, the other driving is stating they had a green left turn arrow.  The investigating officer said none of the witnesses were able to verify either side of the story leaving it at a my word against theirs so no tickets were issued.  I receiver airbag related injuries to my left arm and damage to the front of my car.  I have an older car with liability only insurance and would like the repair cost for my car to be covered by the other driver's insurance.  Even though no tickets were issued do I have a chance at getting my repair costs covered?  All the reading I have done points towards left turning cars being liable in an accident in the absence of proof that I did something wrong, but I'd appreciate some advice before I decide my next steps.

Answer
You are on the right track in your thought process.  Absent of any other proof, the person turning left would generally be considered liable.  The fact that neither of you were ticketed actually has little to do with civil liability.  

If you know deep in your soul that you had a green light, then go ahead and file a claim against the other driver.  Their insurance company may initially deny the claim, forcing you to file a small claims suit, but after that, they will have no real defense in civil court and will likely settle.

Good luck!