The chromatic threshold of a graph H with chromatic number r>=3 can be (r-3)/(r-2), (2r-5)/(2r-3), or (r-2)/(r-1)

You need to know: Graph, isomorphic graphs, subgraph, degree of a vertex of graph G, chromatic number $\chi(G)$ of graph G.

Background: We say that graph G has minimum degree at least M if the degree of each vertex of G is at least M. A graph G is called H-free, if it does not have a subgraph isomorphic to H. The chromatic threshold $\delta_\chi(H)$ of a graph H is the infimum of $d>0$ such that there exists $C=C(H,d)$ for which every H-free graph G with minimum degree at least $d|G|$ satisfies $\chi(G) \leq C$ .

The Theorem: On 8th August 2011, Peter Allen, Julia Böttcher, Simon Griffiths, Yoshiharu Kohayakawa, and Robert Morris submitted to arxiv a paper in which they proved that the chromatic threshold of every graph H with $\chi(H)=r\geq 3$ must be either $\frac{r-3}{r-2}$ , or $\frac{2r-5}{2r-3}$ , or $\frac{r-2}{r-1}$ .

Short context: In 1940th, Zykov and Tutte constructed triangle-free graphs
with arbitrarily large chromatic number. In 1959, Erdős did the same for H-free graphs for any graph H containing a cycle. In 1973, Erdős and Simonovits suggested to investigate this question for graphs with large minimum degree, defined the chromatic threshold $\delta_\chi(H)$ , and asked to determine $\delta_\chi(H)$ for every graph H. The Theorem lists 3 possible values for $\delta_\chi(H)$ for every graph H with $\chi(H)\geq 3$ . The authors also determined exactly for which graphs H $\delta_\chi(H)$ is equal to each value. Together with known fact that $\delta_\chi(H)=0$ if $\chi(H)\leq 2$ , this answers the question of Erdős and Simonovits for all graphs H.