Morrison, Natasha


Weak saturation of the cube in the clique

Author(s): Morrison; Noel

\begin{problem} % Enter your conjecture in LaTeX % You may change "conjecture" to "question" or "problem" if you prefer. Determine $\text{wsat}(K_n,Q_3)$. \end{problem}

Keywords: bootstrap percolation; hypercube; Weak saturation

Extremal $4$-Neighbour Bootstrap Percolation in the Hypercube ★★

Author(s): Morrison; Noel

\begin{problem} Determine the smallest percolating set for the $4$-neighbour bootstrap process in the hypercube. \end{problem}

Keywords: bootstrap percolation; extremal combinatorics; hypercube; percolation

Saturation in the Hypercube ★★

Author(s): Morrison; Noel; Scott

\begin{question} What is the saturation number of cycles of length $2\ell$ in the $d$-dimensional hypercube? \end{question}

Keywords: cycles; hypercube; minimum saturation; saturation

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

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