I had the fuel relay problem years ago. When it was hot in the car (since the relay is under the driver-side dash), the car would crank but not fire (no fuel). Resoldering the relay circuit board fixed that. Five years ago, and again recently, the car would not start when hot (parked for 1-2 hours after use). Both times it was the starter solenoid hanging due to the heat. I could tell it was the starter because when I turned the key from ON to START I could see the dash lights dim, due to the extra current the starter was trying to pull. Replacing the starter fixed that. My Accord is a manual transmission, so in those earlier cases I suspected the clutch switch could be bad. So far it has not been the culprit. On an auto transmission, the switch that determines Park and Neutral could be bad. My Mercedes had that one fail. If you have access to an electrical circuit diagram for your car, trace the path of current from the battery, through the key switch, to the starter, and see what items are in the path that could be bad.