Two-part set systems

The two part Sperner theorem of Katona and Kleitman states that if $X$ is an $n$-element set with partition $X_1 \cup X_2$, and \cF is a family of subsets of $X$ such that no two sets A, B \in \cF satisfy $A \subset B$ (or $B \subset A$) and $A \cap X_i=B \cap X_i$ for some $i$, 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 $X_1$, $X_2$. Along the way, we prove the following new result which may be of independent interest: let \cF, \cG be families of subsets of an $n$-element set such that \cF and \cG are both intersecting and cross-Sperner, meaning that if A \in \cF and B \in \cG, then $A \not\subset B$ and $B \not\subset A$. Then |\cF| +|\cG| < 2^{n-1} and there are exponentially many examples showing that this bound is tight

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

Combinatorics

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Lefschetz Properties for Higher Order Nagata Idealizations

We study a generalization of Nagata idealization for level algebras. These algebras are standard graded Artinian algebras whose Macaulay dual generator is given explicity as a bigraded polynomial of bidegree $(1,d)$. We consider the algebra associated to polynomials of the same type of bidegree $(d_1,d_2)$. We prove that the geometry of the Nagata hypersurface of order $e$ is very similar to the geometry of the original hypersurface. We study the Lefschetz properties for Nagata idealizations of order $e$, proving that WLP holds if $d_1\geq d_2$. We give a complete description of the associated algebra in the monomial square free case.Comment: 16 pages, 4 figures. To appear in Advances in Applied Mathematic