# Noel, Jonathan A.

## Weak saturation of the cube in the clique ★

Author(s): Morrison; Noel

Problem

Determine .

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

Author(s): Morrison; Noel

Problem   Determine the smallest percolating set for the -neighbour bootstrap process in the hypercube.

## Saturation in the Hypercube ★★

Author(s): Morrison; Noel; Scott

Question   What is the saturation number of cycles of length in the -dimensional hypercube?

Keywords: cycles; hypercube; minimum saturation; saturation

## Cycles in Graphs of Large Chromatic Number ★★

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

Conjecture   If , then contains at least cycles of length .

Keywords: chromatic number; cycles

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

Author(s): Morrison; Noel; Scott

Question   Does there exist a constant and a function such that if , then every saturated -Sperner system has cardinality at least ?

## Partitioning the Projective Plane ★★

Author(s): Noel

Throughout this post, by projective plane we mean the set of all lines through the origin in .

Definition   Say that a subset of the projective plane is octahedral if all lines in pass through the closure of two opposite faces of a regular octahedron centered at the origin.
Definition   Say that a subset of the projective plane is weakly octahedral if every set such that is octahedral.
Conjecture   Suppose that the projective plane can be partitioned into four sets, say and such that each set is weakly octahedral. Then each is octahedral.

Keywords: Partitioning; projective plane

## Choosability of Graph Powers ★★

Author(s): Noel

Question  (Noel, 2013)   Does there exist a function such that for every graph , ## Choice Number of k-Chromatic Graphs of Bounded Order ★★

Author(s): Noel

Conjecture   If is a -chromatic graph on at most vertices, then .

## Mixing Circular Colourings ★

Author(s): Brewster; Noel

Question   Is always rational?

Keywords: discrete homotopy; graph colourings; mixing 