Endpoint-deck gluing for one-edge deletion at a 3-uniform Ramsey threshold

Definitions and mathematical statement

For $n\ge4$, let $R=r_3(n,n)$ be the least integer $N$ such that every red--blue coloring of the triples of an $N$-element set has an $n$-element subset whose triples all have the same color. Fix a triple $e\in\binom{[R]}3$, and form the hypergraph $H=K_R^{(3)}-e$, whose edge set is $\binom{[R]}3\setminus\{e\}$. The missing triple receives no color. A monochromatic $K_n^{(3)}$ in $H$ means an $n$-element subset $S\subseteq[R]$ such that all its triples belong to $E(H)$ and have the same color. In particular, $e$ cannot be a subset of $S$.

The question of Dudek and Rödl is whether every red--blue coloring of $E(H)$ necessarily contains such a monochromatic complete hypergraph, for every $n\ge4$. The choice of the deleted edge is irrelevant up to relabeling, but the coloring is otherwise unrestricted. The ordinary Ramsey property of $K_R^{(3)}$ does not immediately answer this: extending a coloring to the missing edge may produce only complete monochromatic sets that use it. The nontrivial range is stated explicitly. If $n=3$, then $r_3(3,3)=3$, deleting the sole edge leaves no $K_3^{(3)}$, and the unqualified assertion fails. This boundary case is excluded rather than silently asserted as part of the conjecture.

Short English statement

At the diagonal triple-system Ramsey threshold, does deleting a single triple preserve the Ramsey property?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20000928 (AIM-COMBINATORICS-0053), 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 6. Original workshop question underlying AIM-COMBINATORICS-0053. https://www.overleaf.com/read/mnvcscjjysvg.