The board / graph theory / #922
Erdős Problem #922
PROVEDdatabase updated 2025-08-31
The statement as posed is true, and a proof exists.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
Let \(k\geq 0\). Let \(G\) be a graph such that every subgraph \(H\) contains an independent set of size \(\geq (n-k)/2\), where \(n\) is the number of vertices of \(H\). Must \(G\) have chromatic number at most \(k+2\)? A question of Erdős and Hajnal [ErHa67b]. The case \(k=0\) is trivial, but they could not prove this even for \(k=1\). This is true, and was proved by Folkman [Fo70b]. See also [73](https://www.erdosproblems.com/73).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved. The authoritative wording is on erdosproblems.com/922.Prizenone
Topicsgraph theory, chromatic number
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/922
Attempts
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