A unifying generalization of Sperner’s theorem
نویسنده
چکیده
Sperner’s bound on the size of an antichain in the lattice P(S) of subsets of a finite set S has been generalized in three different directions: by Erdős to subsets of P(S) in which chains contain at most r elements; by Meshalkin to certain classes of compositions of S; by Griggs, Stahl, and Trotter through replacing the antichains by certain sets of pairs of disjoint elements of P(S). We unify Erdős’s, Meshalkin’s, and Griggs–Stahl–Trotter’s inequalities with a common generalization. We similarly unify their accompanying LYM inequalities. Our bounds do not in general appear to be the best possible.
منابع مشابه
Sperner’s Lemma Implies Kakutani’s Fixed Point Theorem
Kakutani’s fixed point theorem has many applications in economics and game theory. One of its most well-known applications is in John Nash’s paper [8], where the theorem is used to prove the existence of an equilibrium strategy in n-person games. Sperner’s lemma, on the other hand, is a combinatorial result concerning the labelling of the vertices of simplices and their triangulations. It is kn...
متن کاملOriented matroids and Ky Fan's theorem
L. Lovász has shown in [9] that Sperner’s combinatorial lemma admits a generalization involving a matroid defined on the set of vertices of the associated triangulation. We prove that Ky Fan’s theorem admits an oriented matroid generalization of similar nature (Theorem 3.1). Classical Ky Fan’s theorem is obtained as a corollary if the underlying oriented matroid is chosen to be the alternating ...
متن کاملA Discrete Approach to the Poincare-Miranda Theorem
The Poincaré-Miranda Theorem is a topological result about the existence of a zero of a function under particular boundary conditions. In this thesis, we explore proofs of the Poincaré-Miranda Theorem that are discrete in nature that is, they prove a continuous result using an intermediate lemma about discrete objects. We explain a proof by Tkacz and Turzański that proves the Poincaré-Miranda T...
متن کاملGeneralization of Titchmarsh's Theorem for the Dunkl Transform
Using a generalized spherical mean operator, we obtain a generalization of Titchmarsh's theorem for the Dunkl transform for functions satisfying the ('; p)-Dunkl Lipschitz condition in the space Lp(Rd;wl(x)dx), 1 < p 6 2, where wl is a weight function invariant under the action of an associated re ection group.
متن کاملA GENERALIZATION OF A JACOBSON’S COMMUTATIVITY THEOREM
In this paper we study the structure and the commutativity of a ring R, in which for each x,y ? R, there exist two integers depending on x,y such that [x,y]k equals x n or y n.
متن کامل