Let $A$ be a subset of $B = \{1, 2,\ldots,n\}$. Suppose that $F$ is a family of subsets of $B$, each of which is a translation of $A$ and no two of which intersect more than once. What is the maximum size of $F$?
This question is similar to the following question: Choosing subsets of a set with a specified amount of maximum overlap between them
Can we improve the Ray-Chaudhuri-Wilson and Frankl Theorems if we know that each subset is a translation of some given subset?