I came across a contest called the Clarke-Bradbury International Science Fiction Competition (with, I believe, Clarke being author Arthur C. Clarke and Bradbury being Ray Bradbury). It appears to have started in 2002, and is sponsored by the European Space Agency (ESA), with contestants being invited to submits stories and/or art relating to a certain scientific themes. I also found this: "In November 2002, ESA's Technology Transfer and Promotion Office launched the Clarke- Bradbury International Science Fiction Competition as a way to involve young people in thinking about space and become more interested in science and technology:"

https://arc.aiaa.org/doi/abs/1.../6.IAC-03-IAA.8.2.04

I had never heard of this competition before. If anyone has more information about this competition, I hope they will post it.

Do they have to have some sort of license to use the names or can they call it whatever they like?

dandelion, in the left hand column of the article which can be accessed by the link in my above post, it notes that the Competition was endorsed by both Arthur C. Clarke and Ray Bradbury.