A family of subsets of X = {1, ... , n} is said to be t-intersecting if |A B| ≥ t for every A , B ∈ . I (n , t) denotes the set of all such families. I k (n , t) is the subset consisting of those families that contain k-element sets only. Determining the size M (Ik (n, t)) of a largest such family is a problem that goes back over 40 years to a paper [1] of Erdös et al. In the intervening time, a significant amount of literature has developed on this and closely-related problems, and has become highly sophisticated and technical. The paper under review is best suited for an expert reader.
Extending the notation introduced earlier, the cardinality of a largest family in a class of families will be denoted by M( ). The general approach is to express
M( ) in terms of the cardinality of a “canonical” (and more easily counted) family. For example, S(n, t, r) is the t-intersecting family consisting of all subsets of X that contain at least t+r of the first t+2r natural numbers. Sk(n, t, r) is defined analogously. Katona [2] proved the identity M(I(n, t)) = |S(n, t, r)| where r = ⌊(n -t )/2⌋. Ahlswede and Khachatrian [3] found a specific r ≤ ⌊(n - t)/2⌋ such that
M(Ik(n, t)) = Sk(n, t, r). These two results form the starting point for this paper.
The authors are primarily concerned with replacing the “cardinality k” condition with “cardinality ≤ k”. In particular, they are interested in under what conditions on k the identity M(I ≤ k (n, t)) = Sk (n, t, ⌊ (n - t)/2⌋ ) holds. (The obvious extensions of notation have been made.) They give a sufficient condition so that the identity holds for large enough n, but also give a sufficient condition so that it fails for large enough n. These considerations lead them to conjecture that if k < (n + t)/2, then M(I≤ k(n, t)) = |S≤ k(n, t, r)| for a suitable choice of r in the range 0 to ⌊(n - t)/2⌋. This conjecture is supported by asymptotic results.
