Simplex Ramsey numbers and a robust boundary-extension lemma

Definitions and mathematical statement

For $k\ge2$, a $k$-uniform hypergraph is a family of $k$-element subsets of a vertex set. The complete such hypergraph on $k+1$ vertices, $K_{k+1}^{(k)}$, consists of all $k+1$ subsets obtained by omitting one vertex. It is the hypergraph simplex. Let $R_k$ be the least integer $N$ such that, whenever each member of $\binom{[N]}k$ is colored red or blue, there is a set $S\subseteq[N]$ of size $k+1$ for which all members of $\binom Sk$ have the same color. Thus \[ R_k=r_k(k+1,k+1). \] The question is to determine, or sharply estimate, the growth of $R_k$ as $k$ tends to infinity. Both the uniformity and the required clique size grow; the clique has exactly one more vertex than the uniformity. The number of colors remains two throughout.

An upper bound $R_k\le U(k)$ must force a monochromatic simplex in every coloring on $U(k)$ vertices. A lower bound $R_k>L(k)$ requires a coloring on $L(k)$ vertices with no such simplex, meaning every $(k+1)$-vertex set has at least one edge of each color. The desired estimates should clarify the growth scale in $k$, beyond simply applying fixed-uniformity Ramsey finiteness. The vertices carry no order restriction on copies, even though $[N]$ supplies labels.

Short English statement

How rapidly does the two-color Ramsey number of a hypergraph simplex grow with its uniformity?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20000948 (AIM-COMBINATORICS-0073), 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 26. Original workshop question underlying AIM-COMBINATORICS-0073. https://www.overleaf.com/read/mnvcscjjysvg.