If you have a knee injury when you are young does that predispose you to arthritis?

Maybe. Arthritis is damage to the cartilage which covers the end of the bone at the joint. Damage to ligaments, cartilage, or a meniscus (cushion/shock absorber) at a young age will predispose the patient to arthritis as they age. Non-impact aerobic activities and preventing obesity can decrease this risk over time.
Yes. Some people create a cartilage injury at the time of their knee injury when they are young and this can manifest as an arthritic condition decades later.