Intersection properties of subsets of integers (Q5906420)

A family \(\mathcal F\) of subsets of \([1,n]=\{1,2, \ldots, n\}\) is called well-intersecting, if for \(A, B \in {\mathcal F}, A \neq B\), the subset \(A \cap B\) is a non-empty arithmetic progression. The main result of this paper is to show that \(|{\mathcal F}|< n^2/2 + O(n^{5/3}\log^3n)\). A well-intersecting family of cardinality \({n \choose 2} + \left[n-1 \over 4 \right]+1\) is explicitly constructed.