Most people believe that you can only have one kind of relationship status on Facebook- either in a relationship, single, or it's complicated. However, there is a way to change your relationship status to reflect an open relationship, if that is what you are looking for.
To do this, go to your profile page and click on the About tab. Under the Relationships and Family section, there will be a blank field where you can fill in your relationship status. Instead of selecting one of the pre-determined options, select the option that says Customize.
A pop-up window will appear asking you to fill in more details about your relationship. In the first blank field, type in whatever you would like to call your primary partner- this could be something like My Partner. In the second blank field, type in In an Open Relationship. You also have the option of adding a third person to this field, if you so choose.
Once you hit enter or return, your new relationship status will appear on your profile page for all of your friends (and potential partners) to see!