The Frankl-Wilson theorem and some consequences in Ramsey theory and combinatorial geometry

Equality holds if and only if A consists of all r-subsets containing some fixed i ∈ [n]. What happens if, instead of demanding that A ⊂ [n] be intersecting, we demand that |x∩ y| is odd for any two distinct x, y ∈ A? How large can |A| be? It turns out that the answer depends heavily on whether r is even or odd. If r is odd, we can take A to be the family of all r-sets containing 1 and (r − 1)/2 of the pairs {2, 3}, {4, 5}, . . . , {n− 1, n}, giving