Three-uniform hypertrees and Ramsey goodness

Determine whether every 3-uniform tree is n-good for every integer n at least 3, where n-goodness means attaining the standard lower bound for the Ramsey number against the complete 3-uniform hypergraph.

Background

The paper disproves the general conjecture that every r-uniform tree is n-good by exhibiting a 4-uniform tree that is not 5-good. It explicitly notes, however, that the 3-uniform case of the original conjecture remains unresolved.

References

The 3-uniform case of Conjecture 6.2, which is the bulk of Budden & Penland (2017), remains open.

— The Problem Is the Problem: Towards Scalable Mathematical Discovery  (2608.16977 - Zheng et al., 17 Aug 2026) in Remark C.2.2, Appendix C.2

So does their conjecture that any two r-uniform trees of the same order have the same Ramsey number against K(r)n : T is the only 4-uniform tree of order 7.

— The Problem Is the Problem: Towards Scalable Mathematical Discovery  (2608.16977 - Zheng et al., 17 Aug 2026) in Remark C.2.2, Appendix C.2