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If you take a look at the WordPress code base, you’ll see there is similar logic applied to username. The gist will be

1. Query the database for a list of display names
2. Throw an error if the desired name already exists

Hi @david-ar,

consider first https://codex.wordpress.org/Function_Reference/get_currentuserinfo

then read here: https://codex.wordpress.org/Function_Reference/get_currentuserinfo

and finally some snippet, http://wordpress.stackexchange.com/questions/82239/make-display-name-unique

As alternative, it exist a wp plugin called Credible Names

The solution is given in the second stackexchange’s answer !