A New Construction for Cancellative Families of Sets

Following [2], we say a family, H , of subsets of a n-element set is cancellative if A∪B = A∪C implies B = C when A,B,C ∈ H . We show how to construct cancellative families of sets with c2 elements. This improves the previous best bound c2 and falsifies conjectures of Erdös and Katona [3] and Bollobas [1]. AMS Subject Classification. 05C65 We will look at families of subsets of a n-set with the property that A∪B = A∪C ⇒ B = C for any A,B,C in the family. Frankl and Füredi [2] call such families cancellative. We ask how large cancellative families can be. We define f(n) to be the size of the largest possible cancellative family of subsets of a n-set and f(k, n) to be the size of the largest possible cancellative family of k-subsets of a n-set. Note the condition A ∪ B = A ∪ C ⇒ B = C is the same as the condition B4C ⊆ A⇒ B = C where 4 denotes the symmetric difference. Let F1 be a family of subsets of a n1-set, S1. Let F2 be a family of subsets of a n2set, S2. We define the product F1 × F2 to be the family of subsets of the (n1 + n2)-set, S1 ∪ S2, whose members consist of the union of any element of F1 with any element of F2. It is easy to see that the product of two cancellative families is also a cancellative family ((A1, A2)∪ (B1, B2) = (A1, A2)∪(C1, C2)⇒ (A1∪B1, A2∪B2) = (A1∪C1, A2∪ C2) ⇒ A1 ∪ B1 = A1 ∪ C1 and A2 ∪ B2 = A2 ∪ C2 ⇒ B1 = C1 and B2 = C2 ⇒ (B1, B2) = (C1, C2)). Hence f(n1 + n2) ≥ f(n1)f(n2). Similarly f(k1 + k2, n1 + n2) ≥ f(k1, n1)f(k2, n2).