Open Problem Garden

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

Posted by porton
updated January 15th, 2020

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Graph Theory » Extremal G.T.

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

Posted by Jon Noel
updated October 10th, 2019

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density problems

Sidorenko, A.

Graph 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 $\left(\frac{2|E(G)|}{|V(G)|^2}\right)^{|E(H)|}|V(G)|^{|V(H)|}$ .