The minimal empty-core tight path and the 2026 uniformity-four resolution

Definitions and mathematical statement

For integers $k\ge2$ and $s\ge1$, the tight $k$-uniform path with $s$ edges, denoted $P_s^{(k)}$, has vertex set $[s+k-1]$ and edges \[ \{i,i+1,\ldots,i+k-1\},\qquad 1\le i\le s. \] The number $s$ counts edges, not vertices. Write $K_t^{(k)}$ for the complete $k$-uniform hypergraph on $t$ vertices. For $t\ge k$, let $r_k(P_s^{(k)},K_t^{(k)})$ be the least $N$ such that every coloring of the $k$-element subsets of $[N]$ red or blue contains a red copy of $P_s^{(k)}$ or a blue copy of $K_t^{(k)}$. All copies are injective and ordinary, rather than induced, hypergraph copies.

The question is whether for every fixed $k\ge2$ there exists an integer $s_0=s_0(k)$ such that for every fixed $s\ge s_0$, \[ r_k(P_s^{(k)},K_t^{(k)})=\Theta_{k,s}(t^{k-1}) \quad(t\to\infty). \] Explicitly, positive constants $c_{k,s},C_{k,s}$ and a threshold $t_0(k,s)$ must give both corresponding inequalities for all $t\ge t_0(k,s)$. The exponent is prescribed, while the multiplicative constants may depend on the path. Neither $k$ nor $s$ grows with $t$ in this assertion. The central issue is whether making the fixed path sufficiently long forces the full power $t^{k-1}$ in every uniformity. Nie's Theorem 1.1 establishes this order for nontrivial tight trees in uniformities three and four, hence for the corresponding sufficiently long paths. His Conjecture 1 explicitly retains uniformities at least five; these are the remaining range of the general question here.

Short English statement

For every uniformity at least five, do sufficiently long fixed tight paths have clique Ramsey numbers of order the expected full power of the clique size?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20000924 (AIM-COMBINATORICS-0049), 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 2. Original workshop question underlying AIM-COMBINATORICS-0049. https://www.overleaf.com/read/mnvcscjjysvg.
• S3 Jiaxi Nie, On tight tree-complete hypergraph Ramsey numbers, Theorem 1.1 and Conjecture 1. Distinguishes proved uniformities three and four from the higher-uniformity target. https://arxiv.org/pdf/2412.19461.