Loose hypergraph triangles versus cliques: an edge-rooted component certificate

Definitions and mathematical statement

Let $T$ be the $3$-uniform hypergraph with vertex set $[6]$ and edge set \[ \{\{1,2,3\},\{3,4,5\},\{5,6,1\}\}. \] It is a loose triangle: each pair of edges meets in one vertex, and the three intersection vertices are distinct. In particular, it is not the tight three-edge path or a triangle in an ordinary graph. For $t\ge3$, define $R_T(t)=r_3(T,K_t^{(3)})$ as the least $N$ for which every red--blue coloring of $\binom{[N]}3$ contains a red copy of $T$ or a blue complete $3$-uniform hypergraph on $t$ vertices. The red copy uses six distinct vertices and need not be induced.

The proposed improvement is \[ R_T(t)=o(t^{3/2})\qquad(t\to\infty). \] Precisely, for every real $\varepsilon>0$ there should be a threshold $t_0$ such that $R_T(t)\le\varepsilon t^{3/2}$ for all $t\ge t_0$. Equivalently, every sufficiently large $T$-free triple system on $N$ vertices should have an independent set larger than $C N^{2/3}$, for each fixed $C>0$ once $N$ is sufficiently large. An independent set is a set containing no hyperedge. This reformulation uses the red triples as the hypergraph and the blue clique as an independent set. The coefficient is required to improve without bound; a fixed constant improvement to $O(t^{3/2})$ would not suffice.

Short English statement

Is the Ramsey number of a loose triple-system triangle against a clique little-oh of the clique size to the power three halves?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20000949 (AIM-COMBINATORICS-0074), 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 27. Original workshop question underlying AIM-COMBINATORICS-0074. https://www.overleaf.com/read/mnvcscjjysvg.