Problem Statement
The cochromatic number of $G$, denoted by $\zeta(G)$, is the minimum number of colours needed to colour the vertices of $G$ such that each colour class induces either a complete graph or empty graph. Let $\chi(G)$ denote the chromatic number.
If $G$ is a random graph with $n$ vertices and each edge included independently with probability $1/2$ then is it true that almost surely\[\chi(G) - \zeta(G) \to \infty\]as $n\to \infty$?
If $G$ is a random graph with $n$ vertices and each edge included independently with probability $1/2$ then is it true that almost surely\[\chi(G) - \zeta(G) \to \infty\]as $n\to \infty$?
Categories:
Graph Theory Chromatic Number
Progress
A problem of Erdős and Gimbel (see also [Gi16]). At a conference on random graphs in Poznan, Poland (most likely in 1989) Erdős offered \$100 for a proof that this is true, and \$1000 for a proof that this is false (although later told Gimbel that \$1000 was perhaps too much).It is known that almost surely\[\frac{n}{2\log_2n}\leq \zeta(G)\leq \chi(G)\leq (1+o(1))\frac{n}{2\log_2n}.\](The final upper bound is due to Bollobás [Bo88]. The first inequality follows from the fact that almost surely $G$ has clique number and independence number $< 2\log_2n$.)
Heckel [He24] and, independently, Steiner [St24b] have shown that it is not the case that $\chi(G)-\zeta(G)$ is bounded with high probability, and in fact if $\chi(G)-\zeta(G) \leq f(n)$ with high probability then $f(n)\geq n^{1/2-o(1)}$ along an infinite sequence of $n$. Heckel conjectures that, with high probability,\[\chi(G)-\zeta(G) \asymp \frac{n}{(\log n)^3}.\]Heckel [He24c] further proved that, for any $\epsilon>0$, we have\[\chi(G) -\zeta(G) \geq n^{1-\epsilon}\]for roughly $95\%$ of all $n$.
Source: erdosproblems.com/625 | Last verified: January 15, 2026