Sperner partition systems

A \textsl{Sperner $k$-partition system} on a set $X$ is a set of partitions of $X$ into $k$ classes such that the classes of the partitions form a Sperner set system (so no class from a partition is a subset of a class from another partition). These systems were defined by Meagher, Moura and Stevens in \cite{MMS} who showed that if $|X| = k \ell$, then the largest Sperner $k$-partition system has size $\frac{1}{k}\binom{|X|}{\ell}$. In this paper we find bounds on the size of the largest Sperner $k$-partition system where $k$ does not divide the size of $X$, specifically, we give an exact bound when $k=2$ and upper and lower bounds when $|X| = 2k+1$, $|X|=2k+2$ and $|X| = 3k-1$.Comment: 15 page

New bounds on the maximum size of Sperner partition systems

An $(n,k)$-Sperner partition system is a collection of partitions of some $n$-set, each into $k$ nonempty classes, such that no class of any partition is a subset of a class of any other. The maximum number of partitions in an $(n,k)$-Sperner partition system is denoted $\mathrm{SP}(n,k)$. In this paper we introduce a new construction for Sperner partition systems and use it to asymptotically determine $\mathrm{SP}(n,k)$ in many cases as $\frac{n}{k}$ becomes large. We also give a slightly improved upper bound for $\mathrm{SP}(n,k)$ and exhibit an infinite family of parameter sets $(n,k)$ for which this bound is tight.Comment: 20 pages, 2 figure

Disjoint spread systems and fault location

When $k$ factors each taking one of $v$ levels may affect the correctness or performance of a complex system, a test is selected by setting each factor to one of its levels and determining whether the system functions as expected (passes the test) or not (fails). In our setting, each test failure can be attributed to at least one faulty (factor, level) pair. A nonadaptive test suite is a selection of such tests to be executed in parallel. One goal is to minimize the number of tests in a test suite from which we can determine which (factor, level) pairs are faulty, if any. In this paper, we determine the number of tests needed to locate faults when exactly one (or at most one) pair is faulty. To do this, we address an equivalent problem, to determine how many set partitions of a set of size $N$ exist in which each partition contains $v$ classes and no two classes in the partitions are equal.Comment: 16 pages, 0 figure