Cross-Sperner Systems
| dc.contributor.author | Kuperus, Akina | |
| dc.contributor.supervisor | Morrison, Natasha | |
| dc.date.accessioned | 2023-08-22T20:31:36Z | |
| dc.date.available | 2023-08-22T20:31:36Z | |
| dc.date.copyright | 2023 | en_US |
| dc.date.issued | 2023-08-22 | |
| dc.degree.department | Department of Mathematics and Statistics | en_US |
| dc.degree.level | Master of Science M.Sc. | en_US |
| dc.description.abstract | Two sets $A$ and $B$ are \emph{comparable} if $A \subseteq B$ or $B \subseteq A$. A collection of families $(\F_{1}, \F_{2} , \cdots , \F_{k}) \in \mathcal{P}([n])^k$ is \emph{cross-Sperner} if there is no pair $i \not= j$ for which some $F_i \in \F_i$ is comparable to some $F_j \in \F_j$. Two natural measures of the `size' of such systems are the sum $\sum_{i = 1}^k |\F_i|$ and the product $\prod_{i = 1}^k |\F_i|$. Let $\s(n,k)$ be the maximum size of the sum measure for a cross-Sperner system $(\F_{1} , \cdots , \F_{k}) \in \mathcal{P}([n])^k$, and let $\maxprod(n,k)$ be the maximum size of the product measure for a cross-Sperner system $(\F_{1} , \cdots , \F_{k}) \in \mathcal{P}([n])^k$. We prove new upper and lower bounds on $\s(n,k)$ and $\maxprod(n,k)$ for general $n$ and $k \ge 2$ which improve considerably on the previous best bounds. In this thesis we prove that \[\left(\frac{2^n}{ek}\right)^k \le \maxprod(n,k) \le \left(1+\frac{1}{k}\right)\left(\frac{2^n}{2k}\right)^k, \] and \[2^{n} - \frac{3}{\sqrt{2}}\sqrt{2^n k} + 2(k-1) ~\le~ \sigma(n,k) ~\le~ 2^{n} - 2\sqrt{2^n (k-1)} + 2(k-1).\] In particular, we construct a rich family of counterexamples to a conjecture of Gerbner, Lemons, Palmer, Patk\'{o}s, and Sz\'{e}csi from 2011. To prove these bounds, we exploit a connection between cross-Sperner systems and the \emph{comparability number} of a family of sets. Define the comparability number of a family $\F \subseteq \ps{n}$ to be the number of sets comparable to $\F$. Then define $c(n,m)$ to be the minimum comparability number of a family $\F \subseteq \ps{n}$ where $|\F| =m$. We prove that for $1 \leq m \leq 2^n$, \[c(n,m) \geq 2^{n/2 + 1}\sqrt{m} - m.\] This thesis includes joint work with Natasha Morrison, Natalie Behague, and Ashna Wright. | en_US |
| dc.description.scholarlevel | Graduate | en_US |
| dc.identifier.uri | http://hdl.handle.net/1828/15279 | |
| dc.language | English | eng |
| dc.language.iso | en | en_US |
| dc.rights | Available to the World Wide Web | en_US |
| dc.subject | Combinatorics | en_US |
| dc.subject | Extremal Set Theory | en_US |
| dc.subject | Cross-Sperner | en_US |
| dc.subject | Comparability | en_US |
| dc.title | Cross-Sperner Systems | en_US |
| dc.type | Thesis | en_US |