## 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

## Edge-Unfolding Convex Polyhedra ★★

Author(s): Shephard

**Conjecture**Every convex polyhedron has a (nonoverlapping) edge unfolding.

## Singmaster's conjecture ★★

Author(s): Singmaster

**Conjecture**There is a finite upper bound on the multiplicities of entries in Pascal's triangle, other than the number .

The number appears once in Pascal's triangle, appears twice, appears three times, and appears times. There are infinite families of numbers known to appear times. The only number known to appear times is . It is not known whether any number appears more than times. The conjectured upper bound could be ; Singmaster thought it might be or . See Singmaster's conjecture.

Keywords: Pascal's triangle