Restricting a reloid to lattice Gamma before converting it into a funcoid ★★

Author(s): Porton

\begin{conjecture} $(\mathsf{FCD}) f = \bigcap^{\mathsf{FCD}} (\Gamma (A ; B) \cap \operatorname{GR} f)$ for every reloid $f \in \mathsf{RLD} (A ; B)$. \end{conjecture}

Keywords: funcoid corresponding to reloid; funcoids; reloids

Inner reloid through the lattice Gamma ★★

Author(s): Porton

\begin{conjecture} $(\mathsf{RLD})_{\operatorname{in}} f = \bigcap^{\mathsf{RLD}} \operatorname{up}^{\Gamma (\operatorname{Src} f ; \operatorname{Dst} f)} f$ for every funcoid $f$. \end{conjecture}

Counter-example: $(\mathsf{RLD})_{\operatorname{in}} f \sqsubset \bigcap^{\mathsf{RLD}} \operatorname{up}^{\Gamma (\operatorname{Src} f ; \operatorname{Dst} f)} f$ for the funcoid $f = (=)|_\mathbb{R}$ is proved in \href[this online article]{http://www.mathematics21.org/binaries/funcoids-are-filters.pdf}.

Keywords: filters; funcoids; inner reloid; reloids

Chromatic Number of Common Graphs ★★

Author(s): Hatami; Hladký; Kráľ; Norine; Razborov

\begin{question} Do common graphs have bounded chromatic number? \end{question}

Keywords: common graph