3-Edge-Coloring Conjecture ★★★

Author(s): Arthur; Hoffmann-Ostenhof

Conjecture   Suppose $ G $ with $ |V(G)|>2 $ is a connected cubic graph admitting a $ 3 $-edge coloring. Then there is an edge $ e \in E(G) $ such that the cubic graph homeomorphic to $ G-e $ has a $ 3 $-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 $ \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