A Simple Proof of a Theorem of Milner

AbstractA new short proof is given for the following theorem of Milner: An intersecting, inclusion-free family of subsets of ann-element set has at most[formula] members

Intersecting P-free families

We study the problem of determining the size of the largest intersecting P-free family for a given partially ordered set (poset) P. In particular, we find the exact size of the largest intersecting B-free family where B is the butterfly poset and classify the cases of equality. The proof uses a new generalization of the partition method of Griggs, Li and Lu. We also prove generalizations of two well-known inequalities of Bollobás and Greene, Katona and Kleitman in this case. Furthermore, we obtain a general bound on the size of the largest intersecting P-free family, which is sharp for an infinite class of posets originally considered by Burcsi and Nagy, when n is odd. Finally, we give a new proof of the bound on the maximum size of an intersecting k-Sperner family and determine the cases of equality. © 2017 Elsevier Inc

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

We present four main results: (1) A solution to the problem of finding two set systems A and B such that A is r1-intersecting, B is r2-intersecting, A,B are t-cross-intersecting and A+ B is a maximum; (2) A solution to the problem of finding two set systems A and B such that A,B are Sperner, t-cross-intersecting and A+ B is a maximum; (3) A solution to the problem of finding the maximum size of an intersecting set system F such that the complementary degree c( F ) = s for a specified value s; (4) An asymptotic result on the complementary degree of an intersecting set system