Maximal antichains of minimum size

Let $n\geqslant 4$ be a natural number, and let $K$ be a set $K\subseteq [n]:={1,2,...,n}$. We study the problem to find the smallest possible size of a maximal family $\mathcal{A}$ of subsets of $[n]$ such that $\mathcal{A}$ contains only sets whose size is in $K$, and $A\not\subseteq B$ for all ${A,B}\subseteq\mathcal{A}$, i.e. $\mathcal{A}$ is an antichain. We present a general construction of such antichains for sets $K$ containing 2, but not 1. If $3\in K$ our construction asymptotically yields the smallest possible size of such a family, up to an $o(n^2)$ error. We conjecture our construction to be asymptotically optimal also for $3\not\in K$, and we prove a weaker bound for the case $K={2,4}$. Our asymptotic results are straightforward applications of the graph removal lemma to an equivalent reformulation of the problem in extremal graph theory which is interesting in its own right.Comment: fixed faulty argument in Section 2, added reference

Minimizing the regularity of maximal regular antichains of 2- and 3-sets

Let $n\geqslant 3$ be a natural number. We study the problem to find the smallest $r$ such that there is a family $\mathcal{A}$ of 2-subsets and 3-subsets of $[n]=\{1,2,...,n\}$ with the following properties: (1) $\mathcal{A}$ is an antichain, i.e. no member of $\mathcal A$ is a subset of any other member of $\mathcal A$, (2) $\mathcal A$ is maximal, i.e. for every $X\in 2^{[n]}\setminus\mathcal A$ there is an $A\in\mathcal A$ with $X\subseteq A$ or $A\subseteq X$, and (3) $\mathcal A$ is $r$-regular, i.e. every point $x\in[n]$ is contained in exactly $r$ members of $\mathcal A$. We prove lower bounds on $r$, and we describe constructions for regular maximal antichains with small regularity.Comment: 7 pages, updated reference

Minimum Weight Flat Antichains of Subsets

Building on classical theorems of Sperner and Kruskal-Katona, we investigate antichains $\mathcal F$ in the Boolean lattice $B_n$ of all subsets of $[n]:=\{1,2,\dots,n\}$, where $\mathcal F$ is flat, meaning that it contains sets of at most two consecutive sizes, say $\mathcal F=\mathcal{A}\cup\mathcal{B}$, where $\mathcal{A}$ contains only $k$-subsets, while $\mathcal{B}$ contains only $(k-1)$-subsets. Moreover, we assume $\mathcal{A}$ consists of the first $m$ $k$-subsets in squashed (colexicographic) order, while $\mathcal{B}$ consists of all $(k-1)$-subsets not contained in the subsets in $\mathcal{A}$. Given reals $\alpha,\beta>0$, we say the weight of $\mathcal F$ is $\alpha\cdot|\mathcal{A}|+\beta\cdot|\mathcal{B}|$. We characterize the minimum weight antichains $\mathcal F$ for any given $n,k,\alpha,\beta$, and we do the same when in addition $\mathcal F$ is a maximal antichain. We can then derive asymptotic results on both the minimum size and the minimum Lubell function

Sizes of flat maximal antichains of subsets

This is the second of two papers investigating for which positive integers $m$ there exists a maximal antichain of size $m$ in the Boolean lattice $B_n$ (the power set of $[n]:=\{1,2,\dots,n\}$, ordered by inclusion). In the first part, the sizes of maximal antichains have been characterized. Here we provide an alternative construction with the benefit of showing that almost all sizes of maximal antichains can be obtained using antichains containing only $l$-sets and $(l+1)$-sets for some $l$

There is no ODC with all pages isomorphic to ...

Let n be a natural number and C = fP 0 ; : : : ; Pn-1 g a collection of spanning subgraphs of Kn , the complete graph on n vertices. C is called an Orthogonal Double Cover (ODC) if every edge of Kn belongs to exactly two elements of C and every two elements of C have exactly one edge in common. Gronau, Mullin and Schellenberg showed that the complete graph Kn has an ODC whose elements consist of cycles of length at most 4 and an isolated vertex, except for finitely many n. In this paper we scetch the computer aided proof of the nonexistence of such an ODC for n = 11. e--mail: [email protected] y e--mail: [email protected], Research supported in part by NATO grant CRG 940085. 1 Introduction Let K n be the (undirected) complete graph on n vertices and C = fP 0 ; P 1 ; : : : ; P n-1 g a collection of spanning subgraphs of K n , called pages. Throughout this paper we will not distinguish between a graph and its set of edges, if not noticed differently...

Combinatorial Problems Motivated by Databases (2014-02-06)

Uwe Leck of the University of Wisconsin-Superior presents a talk for the Undergraduate Colloquium.Some combinatorial problems and results will be discussed that arise in the context of restoration of lost information in distributed databases. Consider a set T of lattice points in a k x k grid and call it a configuration. You can think of the points in T as faulty nodes that need to be repaired or decoded. Performing a step of decoding means transforming T into a new configuration T' by removing all points belonging to some horizontal or vertical line L, under the constraint that only t points of T are on L (where t is some given number). T is decodable if it can be transformed into the empty set by an appropriate sequence of decoding steps. Examples of interesting questions in this context are: What is the largest size of a decodable configuration? Among all decodable configurations with the same given number of points, which are the hardest to decode (i.e., which require the most decoding steps)? What are the smallest decodable configurations that require some given number of decoding steps? How does all this generalize to higher dimensional grids?UMD Department of Mathematics and Statistic

On Orthogonal Double Covers By Trees

A collection P of n spanning subgraphs of the complete graph Kn is said to be an Orthogonal Double Cover (ODC) if every edge of Kn belongs to exactly two members of P and every two elements of P share exactly one edge. We consider the case when all graphs in P are isomorphic to some tree G and improve former results on the existence of ODCs, especially for trees G of short diameter and for trees G on few vertices

Matrix representations of participation constraints

We discuss the existence of matrix representations for generalized and minimum participation constraints which are frequently used in database design and conceptual modelling. Matrix representations, also known as Armstrong relations, have been studied in literature e.g. for functional dependencies and play an important role in example-based design and for the implication problem of database constraints. The major tool to achieve the results in this paper is a theorem of Hajnal and Szemerédi on the occurrence of clique graphs in a given graph