More precisely, for an arbitrary
undirected graph, let be the family of graphs that do not have as an
induced subgraph. Then, according to the conjecture, there exists a constant such that the -vertex graphs in have either a clique or an independent set of size . In other words, for any
hereditary family of graphs that is not the family of all graphs, there exists a constant such that the -vertex graphs in have either a clique or an independent set of size .
A convenient and symmetric reformulation of the Erdős–Hajnal conjecture is that for every graph , the -free graphs necessarily contain a
perfect induced subgraph of polynomial size. This is because every perfect graph necessarily has either a clique or independent set of size proportional to the square root of their number of vertices, and conversely every clique or independent set is itself perfect.
In contrast, for
random graphs in the
Erdős–Rényi model with edge probability 1/2, both the
maximum clique and the
maximum independent set are much smaller: their size is proportional to the
logarithm of , rather than growing polynomially.
Ramsey's theorem proves that no graph has both its maximum clique size and maximum independent set size smaller than logarithmic. Ramsey's theorem also implies the special case of the Erdős–Hajnal conjecture when itself is a clique or independent set.
Partial results
This conjecture is due to
Paul Erdős and
András Hajnal, who proved it to be true when is a
cograph.[4] They also showed, for arbitrary , that the size of the largest clique or independent set grows superlogarithmically. More precisely, for every there is a constant such that the -vertex -free graphs have cliques or independent sets containing at least vertices.[1][4] The graphs for which the conjecture is true also include those with four verticies or less, all five-vertex graphs,[5][6][7][8] and any graph that can be obtained from these and the cographs by
modular decomposition.[9]
As of 2024, however, the full conjecture has not been proven, and remains an open problem.
An earlier formulation of the conjecture, also by Erdős and Hajnal, concerns the special case when is a 5-vertex
cycle graph.[2] This case has been resolved by
Maria Chudnovsky, Alex Scott,
Paul Seymour, and Sophie Spirkl.[7]
Relation to the chromatic number of tournaments
Alon et al. [9] showed that the following statement concerning
tournaments is equivalent to the Erdős–Hajnal conjecture.
Conjecture. For every tournament , there exists and such that for every -free tournament with vertices .
Here denotes the chromatic number of , which is the smallest such that there is a -coloring for . A coloring of a tournament is a mapping such that the color classes are
transitive for all .
The class of tournaments with the property that every -free tournament has for some constant satisfies this equivalent Erdős–Hajnal conjecture (with ). Such tournaments , called heroes, were considered by Berger et al.[10] There it is proven that a hero has a special structure which is as follows:
Theorem. A tournament is a hero if and only if all its strong components are heroes. A strong tournament with more than one vertex is a hero if and only if it equals or for some hero and some integer .
Here denotes the tournament with the three components , the transitive tournament of size and a single node . The arcs between the three components are defined as follows: . The tournament is defined analogously.