Link structure and sharp edge bounds for three-edge tight-path-free triple systems

Definitions and mathematical statement

A simple $3$-uniform hypergraph has edges that are three-element sets. Let $P_3$ have vertices $1,2,3,4,5$ and edges $\{1,2,3\},\{2,3,4\},\{3,4,5\}$. Thus its length is three edges, and it is a tight path: consecutive edges overlap in two vertices. Let $K_t^{(3)}$ denote the hypergraph containing all triples on $t$ vertices. Define $R(t)=r_3(P_3,K_t^{(3)})$ to be the least $N$ such that every red--blue coloring of the triples of $[N]$ contains either a red copy of $P_3$ or a blue copy of $K_t^{(3)}$. Copies use distinct vertices and need not be induced.

Determine the asymptotic growth of $R(t)$ as $t\to\infty$. The workshop specifically asks whether \[ R(t)=o(t^2), \] and whether the more precise order $\Theta(t^2/\log t)$ is correct. Here the first statement means that $R(t)/t^2$ tends to zero; the second requires positive constant upper and lower multiples for all sufficiently large $t$. Equivalently, one seeks the largest number of vertices of a $P_3$-free triple system whose largest independent set has fewer than $t$ vertices; an independent set contains no edge. This equivalent formulation counts the red edges as the hypergraph. The target concerns exactly three edges, whose intersection contains the middle vertex; results about longer tight paths do not by themselves settle it.

Short English statement

Determine the Ramsey growth of a three-edge tight triple-system path against a large complete triple system.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20000923 (AIM-COMBINATORICS-0048), 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 1. Original workshop question underlying AIM-COMBINATORICS-0048. https://www.overleaf.com/read/mnvcscjjysvg.