Technically the first 3k of the gift (but not an inheritance) from any person in a year is disregarded, I.e. you only consider the amount in excess of 3k.
So in your example, you would pay tax on 7k (assuming no previous use of the lifetime threshold for that category).
If that person together with their spouse for example jointly gave you the gift, you would ignore the first 6k, and only pay tax on 4k.
There's no need to report a gift that avails of the small gift exemption.
You need to file a return as soon as you've used 80% of your lifetime threshold.