A small Fano-good tight path and the connectedness convention

Definitions and mathematical statement

A finite simple $3$-uniform hypergraph consists of a vertex set and a set of three-element subsets, called edges. Let $F$ be the Fano plane: its vertices are the seven nonzero vectors of $\mathbb F_2^3$, and its edges are the unordered triples $\{x,y,x+y\}$ with $x\ne y$ nonzero. These give seven distinct edges. For finite $3$-uniform hypergraphs $H,J$, let $r(H,J)$ be the least $N$ such that every red--blue coloring of the triples of an $N$-element set contains a red copy of $H$ or a blue copy of $J$. A copy is an injective vertex map carrying every required edge to an edge of the designated color; induced containment is not required.

Classify the hypergraphs $H$ for which \[ r(H,F)=2\left(|V(H)|-1\right)+1. \] This equality is the workshop's definition of being Fano-good. The question may be considered for all finite $H$ having at least one edge, including the role of isolated vertices. For connected $H$, the usual lower-bound coloring divides $2(|V(H)|-1)$ vertices into two equal parts, colors triples within a part red, and mixed triples blue. Here connected means connectedness of the vertex--edge incidence graph. That construction's justification uses connectedness of $H$, whereas the numerical equality itself remains a meaningful classification question without it. A solution should identify exactly which structural properties force equality rather than a different Ramsey threshold.

Short English statement

Which three-uniform hypergraphs attain the proposed exact Ramsey threshold against the Fano plane?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20000950 (AIM-COMBINATORICS-0075), AIM Workshop Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 American Institute of Mathematics, Graph Ramsey theory, item 28. Original workshop question underlying AIM-COMBINATORICS-0075. https://www.overleaf.com/read/mnvcscjjysvg.