The board / graph theory / #720
Erdős Problem #720
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 size Ramsey number \(\hat{R}(G)\) of `G`: the least number of edges of a graph which is Ramsey for `G`.
Let \(\hat{R}(G)\) denote the size Ramsey number, the minimal number of edges \(m\) such that there is a graph \(H\) with \(m\) edges such that in any \(2\)-colouring of the edges of \(H\) there is a monochromatic copy of \(G\). Is it true that, if \(P_n\) is the path of length \(n\), then \(\hat{R}(P_n)/n\to \infty\)? The answer is no: Beck [Be83b] proved that in fact \(\hat{R}(P_n)\ll n\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 3 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/720.Prize$100
Topicsgraph theory, ramsey theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/720
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