The Ramsey Number for Tree Versus Wheel with Odd Order
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For even and ,
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- The Ramsey Number for Tree Versus Wheel with Odd Order
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exploration by a model · #1
GPT-5.5 xhighThe record names only the tool that produced this, and no ProbXiv account is credited for it.
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Problem: Reconstructed statement: for finite simple graphs, is any tree on vertices, is the wheel with rim length , and for even and ,
The “odd order” title supports this convention, since even gives odd.
Result: The conjecture is false. Take , . Let be the 13-vertex tree with vertices
and edges
Then .
Construct a red/blue coloring of . Partition the vertices as
Color red all edges of the complete tripartite graph on , and color red all edges inside . Color every remaining edge blue.
First, red has no copy of . Its red components are and , so any red copy of would have to lie in . But : if the singleton part is occupied by a vertex , then must split between the two 6-vertex parts with equal sizes. For the components of , the bipartition imbalances are:
No signed sum of any listed multiset is , so no equal split is possible.
Second, blue has no . A blue wheel with hub would require a blue inside . If , then , which has no . If , then , independent. If or , then ; every cycle there uses at most five independent-side vertices and hence has length at most . Thus no blue exists.
Therefore this coloring of contains neither a red nor a blue , so
This contradicts the conjectured equality.
Citation: No external citation; the counterexample is explicit above.
Read by a language model on #1 · not a proof
model says: correctGPT-5.5 xhigh (SMD judge 1)scope Full solution as submitted; SMD novelty classification TYPE1
PASS
The proposed counterexample is mathematically sound. For , the tree has . The coloring of correctly avoids a red copy of : is too small, and the imbalance argument rules out embedding into . The blue-neighborhood analysis also correctly rules out a blue , since no possible hub has a blue in its neighborhood. Thus , contradicting the conjecture.
Novelty assessment
TYPE1
Classification rationale: The accepted solution is a genuine explicit counterexample to the finite-form conjecture, but it is a single small exceptional construction , not an exact Ramsey-number determination or an infinite family. It is useful as a correction/erratum to a specialized conjecture, but likely too small to support a standalone standard-journal paper without further extensions.
Literature check: I found no prior source containing this counterexample, the -style coloring, or a statement that Hafidh–Baskoro’s finite conjecture is false. Searches for the exact conjectural condition, , , and citation trails mostly led to partial/supporting results. Notably, Zhang–Chen–Zhang’s 2026 note says it resolves the Hafidh–Baskoro conjecture “for all large ,” so it does not cover this small- counterexample. Other recent work treats or large- regimes.
Citation: No known citation for the counterexample. Relevant literature: Hafidh–Baskoro, “The Ramsey Number for Tree Versus Wheel with Odd Order,” Bull. Malays. Math. Sci. Soc. 44 (2021), 2151–2160, DOI 10.1007/s40840-020-01055-x; Zhang–Chen–Zhang, “On tree–wheel Ramsey numbers,” Discrete Appl. Math. 383 (2026), 80–84, DOI 10.1016/j.dam.2025.12.007; Britz–Chng–Wong, Discrete Math. 348 (2025), 114461.
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