40 research outputs found

    On the average rank of LYM-sets

    Get PDF
    Let S be a finite set with some rank function r such that the Whitney numbers wi = |{x S|r(x) = i}| are log-concave. Given so that wk − 1 < wk wk + m, set W = wk + wk + 1 + … + wk + m. Generalizing a theorem of Kleitman and Milner, we prove that every F S with cardinality |F| W has average rank at least kwk + … + (k + m) wk + m/W, provided the normalized profile vector x1, …, xn of F satisfies the following LYM-type inequality: x0 + x1 + … + xn m + 1

    AZ-identities and Strict 2-part Sperner Properties of Product Posets

    Get PDF
    One of the central issues in extremal set theory is Sperner's theorem and its generalizations. Among such generalizations is the best-known BLYM inequality and the Ahlswede--Zhang (AZ) identity which surprisingly generalizes the BLYM inequality into an identity. Sperner's theorem and the BLYM inequality has been also generalized to a wide class of posets. Another direction in this research was the study of more part Sperner systems. In this paper we derive AZ type identities for regular posets. We also characterize all maximum 2-part Sperner systems for a wide class of product posets

    All maximum size two-part Sperner systems - in short

    Get PDF
    In this note we give a very short proof for the description of all maximum size two-part Sperner systems

    Generalized forbidden subposet problems

    Get PDF

    A note on full transversals and mixed orthogonal arrays

    Get PDF
    We investigate a packing problem in M-dimensional grids, where bounds are given for the number of allowed entries in different axis-parallel directions. The concept is motivated from error correcting codes and from more-part Sperner theory. It is also closely related to orthogonal arrays. We prove that some packing always reaches the natural upper bound for its size, and even more, one can partition the grid into such packings, if a necessary divisibility condition holds. We pose some extremal problems on maximum size of packings, such that packings of that size always can be extended to meet the natural upper bound. 1 The concept of full transversals Let us be given positive integers n1,n2,...,nM and L1,L2,...,LM, such tha

    Two-part set systems

    Get PDF
    The two part Sperner theorem of Katona and Kleitman states that if XX is an nn-element set with partition X1∪X2X_1 \cup X_2, and \cF is a family of subsets of XX such that no two sets A, B \in \cF satisfy A⊂BA \subset B (or B⊂AB \subset A) and A∩Xi=B∩XiA \cap X_i=B \cap X_i for some ii, then |\cF| \le {n \choose \lfloor n/2 \rfloor}. We consider variations of this problem by replacing the Sperner property with the intersection property and considering families that satisfiy various combinations of these properties on one or both parts X1X_1, X2X_2. Along the way, we prove the following new result which may be of independent interest: let \cF, \cG be families of subsets of an nn-element set such that \cF and \cG are both intersecting and cross-Sperner, meaning that if A \in \cF and B \in \cG, then A⊄BA \not\subset B and B⊄AB \not\subset A. Then |\cF| +|\cG| < 2^{n-1} and there are exponentially many examples showing that this bound is tight

    The infinite random simplicial complex

    Full text link
    We study the Fraisse limit of the class of all finite simplicial complexes. Whilst the natural model-theoretic setting for this class uses an infinite language, a range of results associated with Fraisse limits of structures for finite languages carry across to this important example. We introduce the notion of a local class, with the class of finite simplicial complexes as an archetypal example, and in this general context prove the existence of a 0-1 law and other basic model-theoretic results. Constraining to the case where all relations are symmetric, we show that every direct limit of finite groups, and every metrizable profinite group, appears as a subgroup of the automorphism group of the Fraisse limit. Finally, for the specific case of simplicial complexes, we show that the geometric realisation is topologically surprisingly simple: despite the combinatorial complexity of the Fraisse limit, its geometric realisation is homeomorphic to the infinite simplex.Comment: 33 page
    corecore