A nonempty subset A of {1, 2, . . . , n} is said to be relatively prime if gcd(A) = 1. Nathanson [4] defined f(n) to be the number of relatively prime subsets of {1, 2, . . . , n} and, for k ≥ 1, fk(n) to be the number of relatively prime subsets of {1, 2, . . . , n} of cardinality k. Nathanson [4] defined Φ(n) to be the number of nonempty subsets A of the set {1, 2, . . . , n} such that gcd(A)...

Let ≽ be a total order on the power set of finite [n]. A subset S ⊂ [n] is separable when for any X,Y and Z − S, ordering X Y same as ∪ Z. The character preference collection all subsets. Motivated by questions in theories voting, marketing social choice, admissibility problem asks which collections \(\mathcal {C} \subset \mathcal {P}({[n]})\) can arise characters orders. We introduce linear al...

One of the basic results in extremal set theory was discovered in [1] and rediscovered in [2]: For a given number of k-element subsets of an n-set the shadow, that is, the set of ( k 1)-element subsets contained in at least one of the specified k-element subsets, is minimal, if the k-element subsets are chosen as an initial segment in the squashed order (see [10]; called colex order in [liD, th...

In European J. Combin. 30 (2009), 986–995, S.M. Hegde recently introduced set colorings of a graph G as an assignment (function) of distinct subsets of a finite set X of colors to the vertices of G, where the colors of the edges are obtained as the symmetric difference of the sets assigned to their end vertices (which are also distinct). A set coloring is called a proper set coloring if all the...

We prove certain Menon-type identities associated with the subsets of set {1, 2,..., n} and related to functions f, f k , Ф defined investigated by Nathanson.

Recently [4] we have obtained some new algebraic results about βN, the Stone-Čech compactification of the discrete set of positive integers and about βW , where W is the free semigroup over a nonempty alphabet with infinitely many variables adjoined. (The results about βW extend the Graham-Rothschild Parameter Sets Theorem.) In this paper we derive some Ramsey Theoretic consequences of these re...

A theorem of Sperner [2] states that a collection of subsets of (l,..., n), no two ordered by inclusion, contains at most ( &,) sets. How many twoelement chains A cB of subsets of {I,..., n} can be found such that sets in different chains are not related? More generally, we seek to determinef,(n), defined to be the maximum m such that there exist subsets A(i,j) 5 {l,..., n}, 1 < i < m, 0 <j < k...