Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
NoIf $\chi(G)$ is finite, ifand all maximal cliques have size at least 3, you may take a graph coloring and unite two colors. This gives a proper coloring of the maximal cliques hypergraph with strictly less than $\chi(G)$ colors.
No, if all maximal cliques have size at least 3, you may take a graph coloring and unite two colors.
If $\chi(G)$ is finite, and all maximal cliques have size at least 3, you may take a graph coloring and unite two colors. This gives a proper coloring of the maximal cliques hypergraph with strictly less than $\chi(G)$ colors.
