## 3-Edge-Coloring Conjecture ★★★

Author(s): Arthur; Hoffmann-Ostenhof

**Conjecture**Suppose with is a connected cubic graph admitting a -edge coloring. Then there is an edge such that the cubic graph homeomorphic to has a -edge coloring.

Keywords: 3-edge coloring; 4-flow; removable edge

## Several ways to apply a (multivalued) multiargument function to a family of filters ★★★

Author(s): Porton

**Problem**Let be an indexed family of filters on sets. Which of the below items are always pairwise equal?

1. The funcoid corresponding to this function (considered as a single argument function on indexed families) applied to the reloidal product of filters .

2. The funcoid corresponding to this function (considered as a single argument function on indexed families) applied to the starred reloidal product of filters .

3. .

Keywords: funcoid; function; multifuncoid; staroid

## Multicolour Erdős--Hajnal Conjecture ★★★

**Conjecture**For every fixed and fixed colouring of with colours, there exists such that every colouring of the edges of contains either vertices whose edges are coloured according to or vertices whose edges are coloured with at most colours.

Keywords: ramsey theory

## Sidorenko's Conjecture ★★★

Author(s): Sidorenko

**Conjecture**For any bipartite graph and graph , the number of homomorphisms from to is at least .

Keywords: density problems; extremal combinatorics; homomorphism