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We study maximal families ${\cal A}$ of subsets of $[n]=\{1,2,\dots,n\}$ such that ${\cal A}$ contains only pairs and triples and $A\not\subseteq B$ for all $\{A,B\}\subseteq{\cal A}$, i.e. ${\cal A}$ is an antichain. For any $n$, all such families ${\cal A}$ of minimum size are determined. This is equivalent to finding all graphs $G=(V,E)$ with $|V|=n$ and with the property that every edge is contained in some triangle and such that $|E|-|T|$ is maximum, where $T$ denotes the set of triangles in $G$. The largest possible value of $|E|-|T|$ turns out to be equal to $\lfloor(n+1)^2/8\rfloor$. Furthermore, if all pairs and triples have weights $w_2$ and $w_3$, respectively, the problem of minimizing the total weight $w({\cal A})$ of ${\cal A}$ is considered. We show that $\min w({\cal A})=(2w_2+w_3)n^2/8+o(n^2)$ for $3/n\leq w_3/w_2=:\lambda=\lambda(n) < 2$. For $\lambda\ge 2$ our problem is equivalent to the (6,3)-problem of Ruzsa and Szemerédi, and by a result of theirs it follows that $\min w({\cal A})=w_2n^2/2+o(n^2)$.

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Keys allow a database management system to uniquely identify tuples in a database. Consequently, the class of keys is of great significance for almost all data processing tasks. In the relational model of data, keys have received considerable interest and are well understood. However, for efficient means of data processing most commercial relational database systems deviate from the relational model. For example, tuples may contain only partial information in the sense that they contain so-called null values to represent incomplete information. Codd's principle of entity integrity says that every tuple of every relation must not contain a null value on any attribute of the primary key. Therefore, a key over partial relations enforces both uniqueness and totality of tuples on the attributes of the key. On the basis of these two requirements, we study the resulting class of keys over relations that permit occurrences of Zaniolo's null value 'no-information'. We show that the interaction of this class of keys is different from the interaction of the class of keys over total relations. We establish a finite ground axiomatization, and an algorithm for deciding the associated implication problem in linear time. Further, we characterize Armstrong relations for an arbitrarily given sets of keys; that is, we give a sufficient and necessary condition for a partial relation to satisfy a key precisely when it is implied by a given set of keys. We also establish an algorithm that computes an Armstrong relation for an arbitrarily given set of keys. While the problem of finding an Armstrong relation for a given key set is precisely exponential in general, our algorithm returns an Armstrong relation whose size is at most quadratic in the size of a minimal Armstrong relation. Finally, we settle various questions related to the maximal size of a family of non-redundant key sets. Our results help to bridge the gap between the existing theory of database constraints and database practice.

Macaulay posets are posets for which there is an analogue of the classical Kruskal-Katona theorem for finite sets. These posets are of great importance in many branches of combinatorics and have numerous applications. We survey mostly new and also some old results on Macaulay posets, where the intention is to present them as pieces of a general theory. In particular, the classical examples of Macaulay posets are included as well as new ones. Emphasis is also put on the construction of Macaulay posets, and their relations to other discrete optimization problems.