What is the relationship between San Xiao and the Seven Fairies?

There is no relationship between the two.

"Three Empresses in the Cloud" is the role in the novel Romance of Gods written by Xu in Ming Dynasty, and "Seven Fairies" is the heroine in the 2007 TV series "The Fairy Couple", so there is no relationship between them.

"Empress San Xiao" refers to three fairies practicing in Sanxian Island, and "Seven Fairies" sometimes refers to the youngest one in Seven Sisters.