## 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

## The Double Cap Conjecture ★★

Author(s): Kalai

**Conjecture**The largest measure of a Lebesgue measurable subset of the unit sphere of containing no pair of orthogonal vectors is attained by two open caps of geodesic radius around the north and south poles.

Keywords: combinatorial geometry; independent set; orthogonality; projective plane; sphere

## Circular flow numbers of $r$-graphs ★★

Author(s): Steffen

A nowhere-zero -flow on is an orientation of together with a function from the edge set of into the real numbers such that , for all , and .

A -regular graph is a -graph if for every with odd.

**Conjecture**Let be an integer. If is a -graph, then .

Keywords: flow conjectures; nowhere-zero flows

## Circular flow number of regular class 1 graphs ★★

Author(s): Steffen

A nowhere-zero -flow on is an orientation of together with a function from the edge set of into the real numbers such that , for all , and . The circular flow number of is inf has a nowhere-zero -flow , and it is denoted by .

A graph with maximum vertex degree is a class 1 graph if its edge chromatic number is .

**Conjecture**Let be an integer and a -regular graph. If is a class 1 graph, then .