## Multicolour Erdős--Hajnal Conjecture ★★★

Author(s): Erdos; Hajnal

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 .

## Edge-Unfolding Convex Polyhedra ★★

Author(s): Shephard

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

Keywords: folding; nets

## 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