Hi @Binal_Patel,
To be clear - are you wanting to check if the linked record has not been linked to any other record in the “Event” table? Or do you want the name of the linked record to be a unique value (but still be available to be linked to multiple “Event” records)?
The latter is easy to solve by appending a unique identifier such as the record id or the value of an auto-number field to a descriptive name using a formula field as your primary field. This would ensure no two records in that table have the same primary key. For example use a formula like this as your primary field and combine two other fields:
{record name} & "-" & {autonumber field name}
The former is a little more complex to solve but can be done using an automation script that is triggered on a change to your linked field. This would require some scripting knowledge.
You can set up a view that will filter out anyone who has already registered. Then limit registration to that view.