Erdős–Ko–Rado with conditions on the minimum complementary degree

Let X = { 1 , 2 , … , n } , 2 k < n , and let X ( k ) denote the set of all subsets of X of size k . A set system F ⊆ X ( k ) is intersecting if no two of its elements are disjoint. The minimum complementary degree c ( F ) of F is the minimum over all i ∈ X of the number of sets in F not containing i . A set system F ⊆ X ( k ) is complementary degree condition maximal (CDCM) if H ⊆ X ( k ) intersecting, c ( H ) ⩾ c ( F ) ⇒ | H | ⩽ | F | and SCDCM (S for strict) if equality holds on the right only if it holds on the left. In this paper we characterize all intersecting F ⊆ X ( k ) with c ( F ) ⩽ n - 3 k - 2 which are CDCM and SCDCM. The characterization is in terms of the cascade form of c ( F ) , a representation in terms of sums of certain binomial coefficients. This result generalizes the Erdős–Ko–Rado and Hilton–Milner theorems and a theorem of Franklʹs with conditions on the maximum degree. The number of isomorphically distinct set systems F ⊆ X ( k ) which are SCDCM is the Catalan number C k - 2 = 1 k - 1 2 k - 4 k - 2 , and the number which are CDCM is ∑ i = 0 k - 2 C i .