Erdős–Hajnal large-girth subgraph conjecture

Establish that for every pair of integers k ≥ 0 and ℓ ≥ 3, there exists an integer f(k,ℓ) such that every graph G with chromatic number at least f(k,ℓ) contains a subgraph H with chromatic number greater than k and girth greater than ℓ.

Background

The paper places this conjecture in the classical graph-theoretic setting. Erdős established the existence of graphs having arbitrarily large girth and chromatic number, and Erdős and Hajnal subsequently conjectured that every graph of sufficiently large chromatic number contains a subgraph that simultaneously retains chromatic number above any prescribed threshold and has arbitrarily large girth.

The paper notes that only limited cases are known: Rödl proved the case ℓ = 3, while other cited work provides partial results and lower bounds. The conjecture is also used as a benchmark for the directed tournament analogue developed in the paper.

References

A few years later, he and Hajnal conjectured that moreover, every graph with large enough chromatic number contains a subgraph with large girth and large chromatic number.

— Digraphs of Large Girth and Dichromatic Number in Tournaments with Large Dichromatic Number  (2609.21895 - Charbit et al., 18 Sep 2026) in Section 1, immediately before Conjecture 1 (Conjecture [Erd–Hajnal])

We further conjecture that the subdigraph H can be found with no small cycles at all.

— Digraphs of Large Girth and Dichromatic Number in Tournaments with Large Dichromatic Number  (2609.21895 - Charbit et al., 18 Sep 2026) in Section 1, immediately before Conjecture 2; discussed further in Section 4, "On Conjecture 2"