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

Author(s): Porton

Problem   Let $ \mathcal{X} $ 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 $ \mathcal{X} $.

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

3. $ \bigcap_{F\in\operatorname{up}^{\mathrm{FCD}}\prod^{\mathrm{Strd}}\mathcal{X}}\langle f \rangle F $.

Keywords: funcoid; function; multifuncoid; staroid

Multicolour Erdős--Hajnal Conjecture ★★★

Author(s): Erdos; Hajnal

Conjecture   For every fixed $ k\geq2 $ and fixed colouring $ \chi $ of $ E(K_k) $ with $ m $ colours, there exists $ \varepsilon>0 $ such that every colouring of the edges of $ K_n $ contains either $ k $ vertices whose edges are coloured according to $ \chi $ or $ n^\varepsilon $ vertices whose edges are coloured with at most $ m-1 $ colours.

Keywords: ramsey theory

Sidorenko's Conjecture ★★★

Author(s): Sidorenko

Conjecture   For any bipartite graph $ H $ and graph $ G $, the number of homomorphisms from $ H $ to $ G $ is at least $ \left(\frac{2|E(G)|}{|V(G)|^2}\right)^{|E(H)|}|V(G)|^{|V(H)|} $.

Keywords: density problems; extremal combinatorics; homomorphism

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 $ 1 $.

The number $ 2 $ appears once in Pascal's triangle, $ 3 $ appears twice, $ 6 $ appears three times, and $ 10 $ appears $ 4 $ times. There are infinite families of numbers known to appear $ 6 $ times. The only number known to appear $ 8 $ times is $ 3003 $. It is not known whether any number appears more than $ 8 $ times. The conjectured upper bound could be $ 8 $; Singmaster thought it might be $ 10 $ or $ 12 $. See Singmaster's conjecture.

Keywords: Pascal's triangle