
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