Noel, Jonathan A.

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

Cycles in Graphs of Large Chromatic Number ★★

Author(s): Brewster; McGuinness; Moore; Noel

\begin{conjecture} If $\chi(G)>k$, then $G$ contains at least $\frac{(k+1)(k-1)!}{2}$ cycles of length $0\bmod k$. \end{conjecture}

Keywords: chromatic number; cycles

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

Partitioning the Projective Plane ★★

Author(s): Noel

Throughout this post, by \emph{projective plane} we mean the set of all lines through the origin in $\mathbb{R}^3$.

\begin{definition} Say that a subset $S$ of the projective plane is \emph{octahedral} if all lines in $S$ pass through the closure of two opposite faces of a regular octahedron centered at the origin. \end{definition}

\begin{definition} Say that a subset $S$ of the projective plane is \emph{weakly octahedral} if every set $S'\subseteq S$ such that $|S'|=3$ is octahedral. \end{definition}

\begin{conjecture} Suppose that the projective plane can be partitioned into four sets, say $S_1,S_2,S_3$ and $S_4$ such that each set $S_i$ is weakly octahedral. Then each $S_i$ is octahedral. \end{conjecture}

Keywords: Partitioning; projective plane

Choosability of Graph Powers ★★

Author(s): Noel

\begin{question}[Noel, 2013] Does there exist a function $f(k)=o(k^2)$ such that for every graph $G$, \[\text{ch}\left(G^2\right)\leq f\left(\chi\left(G^2\right)\right)?\] \end{question}

Keywords: choosability; chromatic number; list coloring; square of a graph

Choice Number of k-Chromatic Graphs of Bounded Order ★★

Author(s): Noel

\begin{conjecture} If $G$ is a $k$-chromatic graph on at most $mk$ vertices, then $\text{ch}(G)\leq \text{ch}(K_{m*k})$. \end{conjecture}

Keywords: choosability; complete multipartite graph; list coloring

Mixing Circular Colourings

Author(s): Brewster; Noel

\begin{question} Is $\mathfrak{M}_c(G)$ always rational? \end{question}

Keywords: discrete homotopy; graph colourings; mixing

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