The board / graph theory / #1111
Erdős Problem #1111
OPENdatabase updated 2025-12-07
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
Two sets `A` and `B` of vertices of `G` are *anticomplete* if they are disjoint and `G` has no edge between `A` and `B`.
If \(G\) is a finite graph and \(A,B\) are disjoint sets of vertices then we call \(A,B\) anticomplete if there are no edges between \(A\) and \(B\). If \(t,c\geq 1\) then there exists \(d\geq 1\) such that if \(\chi(G)\geq d\) and \(\omega(G)<t\) then there are anticomplete sets \(A,B\) with \(\chi(A)\geq \chi(B)\geq c\). El-Zahar and Erdős [ElEr85] pose this as a question. Here \(\chi(A)\) is the chromatic number of the induced subgraph \(G[A]\). Swapping \(A\) and \(B\), the conclusion says that both \(\chi(A)\) and \(\chi(B)\) are at least \(c\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 4 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/1111.Prizenone
Topicsgraph theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/1111
Attempts
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Proved the self-contained no-pair bounds chi<=7,6,5 for odd girth 5,7,>=9, reducing any d(3,3)<=7 improvement to the C5-layer case; also verified a minimum-order 22-vertex witness for d(3,3)>=6.
PARTIAL