Cyclic and dihedral Ramsey numbers of alternating paths versus cliques
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Are the dihedral and cyclic Ramsey numbers for an alternating three-vertex path versus the clique K_b both 2b−1 for every integer b≥1? The symmetry restriction is part of each Ramsey problem.
Why it matters and what counts as progress
Why it matters: The formulas would determine an exact Ramsey case under two different symmetry restrictions, rather than just a general path–clique bound.
What counts as progress: Match a construction giving the lower bound with an upper bound valid in each symmetry class.
How the problem developed
- 17 Apr 2026 · Computational results [1]
Nino Bašić, Ivan Damnjanović, Dragan Stevanović and Ivan Stošić Introduced cyclic Ramsey numbers and used SAT computations to determine several small ordered and cyclic Ramsey numbers, including alternating-path cases.
- 7 Jul 2026 · Computational study [2]
Ivan Damnjanović and Irena Đorđević Studied reflective and dihedral Ramsey numbers with a SAT-based method and recorded a formula conjecture for dihedral alternating paths versus cliques.
- 13 Aug 2026 · Reported proof [3]
Alternating Path Ramsey project repository Posted a proof and partial Lean formalization for the dihedral formula when the alternating path parameter a is at least 4.
AI contribution
The a = 3 slice of two permutational Ramsey conjectures: R_dih(P3alt, K_b) = R_cyc(P3alt, K_b) = 2b − 1 for all b
For every integer b≥1, the dihedral and cyclic Ramsey numbers of the alternating 3-path versus K_b both equal 2b−1.
Sources
- Bašić et al., Some results on small ordered and cyclic Ramsey numbers
- Damnjanović and Đorđević, Computation of small reflective and dihedral Ramsey numbers
- Alternating Path Ramsey project: dihedral formula evidence and partial Lean formalization
- Alternating 3-path Ramsey theorem preprint
- Dihedral Ramsey project verification summary
- Dihedral Ramsey verification map
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Publication dates and editorial changes.