The phrase "only criteria" means that there is only one thing that is being used to make a decision or judgement. It is the sole factor that is being considered in a situation.
I am constantly baffled by people whose only criterion for entering into the relationship is «I love him / her». (marriage.com)
If you have not met your special woman, taking your search to Russia will expose you to several options that meet not only your criteria but also have much more to offer. (russian-dating-secrets.com)
A semi-custom rubric that includes only the criteria selected! (educationworld.com)