Cross-Sperner Systems




Kuperus, Akina

Journal Title

Journal ISSN

Volume Title



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.



Combinatorics, Extremal Set Theory, Cross-Sperner, Comparability