The board / graph theory / #745
Erdős Problem #745
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).
The edge probability \(1/n\), with zero at the empty index.
The independent-edge random graph \(G(n,1/n)\).
The multiset of component sizes, retaining repetitions.
In \(G(n,1/n)\) the second-largest component has order \(n^{2/3}\) in probability. The two positive constants may depend on the probability tolerance.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 2 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/745.Prizenone
Topicsgraph theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/745
Attempts
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