A generalization of the Erdös-Ko-Rado theorem

In this note, we investigate some properties of local Kneser graphs defined in [János Körner, Concetta Pilotto, Gábor Simonyi, Local chromatic number and sperner capacity, J. Combin. Theory Ser. B 95 (1) (2005) 101-117]. In this regard, as a generalization of the Erdös-Ko-Rado theorem, we characterize the maximum independent sets of local Kneser graphs. Next, we provide an upper bound for their chromatic number.     

A counterexample to the generalization of Sperner's theorem

It has been conjectured that the analog of Sperner's theorem on non-comparable subsets of a set holds for arbitrary geometric lattices, namely, that the maximal number of non-comparable elements in a finite geometric lattice is max w(k), where w(k) is the number of elements of rank k. It is shown in this note that the conjecture is not true in general. A class of geometric lattices, each of which is a bond lattice of a finite graph, is constructed in which the conjecture fails to hold.     

A generalization of Dirichlet's unit theorem

In this paper, we obtain a unit theorem for algebraic tori defined over an algebraic number field, which generalizes Dirichlet's unit theorem as well as the S-unit theorem due to Hasse and Chevalley.     

A generalization of steinhagen’s theorem

The theorem of Steinhagen establishes a relation between inradius and width of a convex set. The half of the width can be interpreted as the minimum of inradii of all 1-dimensional orthogonal projections of a convex set. By considering i-dimensional projections we obtain series ofi-dimensional inradii. In this paper we study some relations between these inradii and by this we find a natural generalization of Steinhagen’s theorem. Further we show in the course of our investigation that the minimal error of the triangle inequality for a set of vectors cannot be too large.     

A generalization of dirac’s theorem

LetG be an (r+2)-connected graph in which every vertex has degree at leastd and which has at least 2d-r vertices. Then, for any pathQ of lengthr and vertexy not onQ, there is a cycle of length at least 2d-r containing bothQ andy.     

On a generalization of Pólya’s theorem

We consider generalizations of Pólya’s theorem to the case of entire functions taking algebraic values at natural or integer points.     

A generalization of Browder’s fixed point theorem with applications

In this paper, the concept of c$c$-compact mapping is introduced. A generalization of Browder’s fixed point theorem and some equivalence forms are given. As applications, the existence of solutions for some variational inequalities and monotone operator equations is discussed.     

A generalization of Ramsey's theorem for regular trees

An analog of Ramsey's theorem for regular trees is proved. The original theorem is a special case of it. It is shown that the denumerable case of Ramsey's theorem has no analog for binary trees.     

A simple proof of the Erdös-Chao Ko-Rado theorem

A simple proof is given of the theorem of Erdös, Chao Ko, and Rado.     

A generalization of Krasnoselskii’s eigenvalue theorem

We give a generalization of Krasnoselskii’s eigenvalue theorem to countably condensing set-valued maps in Banach spaces, where the method is to use a fixed point theorem for compact maps. This is based on the fact that there is a compact fundamental set for a countably condensing map.     

On a generalization of Jentzsch’s theorem

Let E be a compact subset of with connected, regular complement and let G(z) denote Green’s function of Ω with pole at ∞. For a sequence (p_n)_nΛ of polynomials with degp_n=n, we investigate the value-distribution of p_n in a neighbourhood U of a boundary point z₀ of E if G(z) is an exact harmonic majorant of the subharmonic functions

in . The result holds for partial sums of power series, best polynomial approximations, maximally convergent polynomials and can be extended to rational functions with a bounded number of poles.     

A generalization of Sperner’s theorem and an application to graph orientations

A generalization of Sperner’s theorem is established: For a multifamily M={Y₁,…,Y_p} of subsets of {1,…,n} in which the repetition of subsets is allowed, a sharp lower bound for the number φ(M) of ordered pairs (i,j) satisfying i≠j and Y_i⊆Y_j is determined. As an application, the minimum average distance of orientations of complete bipartite graphs is determined.     

A GENERALIZATION OF CLARKE＇S FIXED POINT THEOREM

This paper proposes a formally stronger set-valued Claxke‘s fixed point theorem. By this-theorem we can improve a fixed point theorem for weakly inward contraction set-valued mapping of D. Dowing and W.A. Kirk.     

A generalization of Dahlberg's theorem concerning the regularity of harmonic Green potentials

Let be the solution operator for in , Tr on , where is a bounded domain in . B. E. J. Dahlberg proved that for a bounded Lipschitz domain maps boundedly into weak- and that there exists such that is bounded for . In this paper, we generalize this result by addressing two aspects. First we are also able to treat the solution operator corresponding to Neumann boundary conditions and, second, we prove mapping properties for these operators acting on Sobolev (rather than Lebesgue) spaces.