The board / graph theory / #87
Erdős Problem #87
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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Let \(0 < \epsilon < 1\). Is it true that, if \(k\) is sufficiently large, then \[R(G) > (1-\epsilon)^k R(k)\] for every graph \(G\) with chromatic number \(\chi(G)=k\)? The restriction \(\epsilon < 1\) excludes negative bases in \((1-\epsilon)^k\). This problem is #12 in Ramsey Theory in the graphs problem collection.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/87.Prizenone
Topicsgraph theory, ramsey theory
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA059442
Original pageerdosproblems.com/87
Attempts
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proved the first inequality for every epsilon > 0.627760438360797, reduced the k=5 lower bound 43 to 4,195 checked critical candidates, and certified R(K_2 join C_5) >= 26; both full asymptotic questi
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