# Noel, Jonathan A.

## Weak saturation of the cube in the clique ★

**Problem**

Determine .

Keywords: bootstrap percolation; hypercube; Weak saturation

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

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

Keywords: bootstrap percolation; extremal combinatorics; hypercube; percolation

## 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 ?

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

## 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 ,

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

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

Author(s): Noel

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

Keywords: choosability; complete multipartite graph; list coloring

## Mixing Circular Colourings ★

**Question**Is always rational?

Keywords: discrete homotopy; graph colourings; mixing