In the set cover problem we are given a collection of m sets whose union covers [n] = {1, . . . , n} and must find a minimum-sized subcollection whose union still covers [n]. We investigate the approximability of set cover by an approximation ratio that depends only on m and observe that, for any constant c < 1/2, set cover cannot be approximated to within O(2 1−1/(log log m)c m) unless SAT can...

The general theme will be the use of model-theoretic methods in proofs of the Manin-Mumford conjecture and variants. If K is an algebraically closed field, G a commutative algebraic group over K and Γ an abstract subgroup of G(K), we will say that Γ is of Lang type, if for any n and subvariety X of G, X(K) ∩ Γ is a finite union of cosets of subgroups of Γ. The Manin-Mumford conjecture says that...

‎In this paper, a new class of fuzzy sets called fuzzy strongly ${{g}^{*}}$-closed sets is introduced and its properties are investigated. Moreover, we study some more properties of this type of closed spaces.

Let x(G) and w(G) denote the chromatic number and clique number (maximum size of a clique) of a graph G. To avoid trivial cases, we always assume that w (G);?: 2. It is well known that interval graphs are perfect, in particular x( G)= w (G) for every interval graph G. In this paper we study the closeness of x and w for two well-known non-perfect relatives of interval graphs: multiple interval g...

The degree structure of complete sets under 2DFA reductions is investigated. It is shown that, for any class C that is closed under log-lin reductions: • All complete sets for the class C under 2DFA reductions are also complete under one-one, length-increasing 2DFA reductions and are first-order isomorphic. • The 2DFA-isomorphism conjecture is false, i.e., the complete sets under 2DFA reduction...

A family of sets in the plane is said to have a 3-transversal if there exists a set of 3 points such that each member of the family contains at least one of them. A conjecture of Grünbaum’s says that a planar family of translates of a convex compact set has a 3-transversal provided that any two of its members intersect. Recently the conjecture has been proved affirmatively (see [1]). We provide...

We investigate some aspects of proof methods for the termination of (extensions of) the second-order λ-calculus in presence of union and existential types. We prove that Girard’s reducibility candidates are stable by union iff they are exactly the non-empty sets of terminating terms which are downward-closed w.r.t. a weak observational preorder. We show that this is the case for the Curry-style...

Introduction LET dim-X be the covering dimension of a space X and let ind X and Ind X be the dimensions defined inductively in terms of the boundaries of neighbourhoods of points and closed sets respectively. The local dimension loo dim X is the least number n such that every point has a closed neighbourhood U with dim U ^ n. The local inductive dimension locIndX is defined analogously, while i...

A family $F$ of $k$-sets on an $n$-set $X$ is said to be $(s,t)$-union intersecting if for any $A_1,\ldots,A_{s+t}$ in this family, we have $\left(\cup_{i=1}^s A_i\right)\cap\left(\cup_{i=1}^t A_{i+s}\right)\neq \varnothing.$ The celebrated Erdős-Ko-Rado theorem determines the size and structure largest (or $(1,1)$-union intersecting) family. Also, Hilton-Milner second $k$-sets. In paper, $t\ge...