Bounds on maximal families of sets not containing three sets with A ∩ B ⊂ C , A 6 ⊂ B

Functionally closed sets and functionally convex sets in real Banach spaces

‎Let $X$ be a real normed  space, then  $C(subseteq X)$  is  functionally  convex  (briefly, $F$-convex), if  $T(C)subseteq Bbb R $ is  convex for all bounded linear transformations $Tin B(X,R)$; and $K(subseteq X)$  is  functionally   closed (briefly, $F$-closed), if  $T(K)subseteq Bbb R $ is  closed  for all bounded linear transformations $Tin B(X,R)$. We improve the    Krein-Milman theorem  ...

FC-families and improved bounds for Frankl's conjecture

A family of sets A is said to be union-closed if {A∪B : A, B ∈ A} ⊂ A. Frankl’s conjecture states that given any finite union-closed family of sets, not all empty, there exists an element contained in at least half of the sets. Here we prove that the conjecture holds for families containing three 3-subsets of a 5-set, four 3-subsets of a 6-set, or eight 4-subsets of a 6-set, extending work of P...

Zero sets in pointfree topology and strongly $z$-ideals

In this paper a particular case of z-ideals, called strongly z-ideal, is defined by introducing zero sets in pointfree topology. We study strongly z-ideals, their relation with z-ideals and the role of spatiality in this relation. For strongly z-ideals, we analyze prime ideals using the concept of zero sets. Moreover, it is proven that the intersection of all zero sets of a prime ideal of C(L),...

On bD-Sets and Associated Separation Axioms

On the Density of Integral Sets with Missing Differences

For a given set M of positive integers, a well-known problem of Motzkin asks for determining the maximal density μ(M) among sets of nonnegative integers in which no two elements differ by M . The problem is completely settled when |M | ≤ 2, and some partial results are known for several families of M for |M | ≥ 3. In this paper, we consider the case M = {a, b, c}, with c a multiple of a or b. I...