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

Abstract

2, . . . , n}, 2k < 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.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 Frankl's 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 2k−4 k−2 , and the number which are CDCM is k−2 i=0 C i .

Maximum size t-cross-intersecting and intersecting families with degree conditions

“…This theorem has turned out to be a key ingredient in the proofs of many extremal theorems, including Katona's first proof of the Erdős-Ko-Rado theorem and his proof of the maximum tintersecting set system theorem [13], Milner's proof of the maximum size of a t-intersecting Sperner family [26] (Theorem 3A in this chapter), and Frankl's [6] and Goldwasser's [9] proofs of theorems on the maximum size of an intersecting set system with conditions on the degrees.…”