40 research outputs found
On the average rank of LYM-sets
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
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
In this note we give a very short proof for the description of all maximum size two-part Sperner systems
A note on full transversals and mixed orthogonal arrays
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
The two part Sperner theorem of Katona and Kleitman states that if is an
-element set with partition , and \cF is a family of subsets
of such that no two sets A, B \in \cF satisfy (or ) and for some , 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 , . Along the way, we prove the following new result which may
be of independent interest: let \cF, \cG be families of subsets of an
-element set such that \cF and \cG are both intersecting and
cross-Sperner, meaning that if A \in \cF and B \in \cG, then and . Then |\cF| +|\cG| < 2^{n-1} and there are
exponentially many examples showing that this bound is tight
The infinite random simplicial complex
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