Saturated $k$-Sperner Systems of Minimum Size ★★

Author(s): Morrison; Noel; Scott

\begin{question} Does there exist a constant $c>1/2$ and a function $n_0(k)$ such that if $|X|\geq n_0(k)$, then every saturated $k$-Sperner system $\mathcal{F}\subseteq \mathcal{P}(X)$ has cardinality at least $2^{(1+o(1))ck}$? \end{question}

Keywords: antichain; extremal combinatorics; minimum saturation; saturation; Sperner system

Antichains in the cycle continuous order ★★

Author(s): DeVos

If $G$,$H$ are graphs, a function $f : E(G) \rightarrow E(H)$ is called \emph{cycle-continuous} if the pre-image of every element of the (binary) cycle space of $H$ is a member of the cycle space of $G$.

\begin{problem} Does there exist an infinite set of graphs $\{G_1,G_2,\ldots \}$ so that there is no cycle continuous mapping between $G_i$ and $G_j$ whenever $i \neq j$ ? \end{problem}

Keywords: antichain; cycle; poset

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