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Erdős Problem #556
DECIDABLEdatabase updated 2025-08-31
Still open, but reduced to a finite computation: checking enough cases would settle it either way.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
Erdős Problem 556 (Bondy–Erdős): for all \(n \ge 4\), \[R(C_n; 3) \le 4n - 3.\] The case \(n = 3\) is excluded since \(R(K_3; 3) = 17\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 5 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/556.Prizenone
Topicsgraph theory, ramsey theory
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA389335
Original pageerdosproblems.com/556
Attempts
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The live statement is false at n=3 (R_3(C_3)=17>9); for the intended n>=4 problem, R_3(C_4)=11 and the sharp Toeplitz K32/K33 C_9 transition are verified, while unrestricted R_3(C_9)<=33 remains open.
FOUND