The hypergraph connector extension of the Cockayne–Lorimer theorem

Let $q,s\ge 1$ and $r\ge 2$ be integers. An $s$-connector is an $r$-uniform hypergraph such that every collection of pairwise disjoint vertex sets $A_1,\dots,A_r$ with $|A_i|\ge s$ contains an edge with one vertex in each $A_i$. For $\mathbf{t}\in\mathbb{N}_+^q$, write $\|\mathbf{t}\|_\infty=\max_i t_i$ and let $\Lambda_{\mathbf{t}}$ denote the corresponding Ramsey threshold term. There exists $\varphi=\varphi_{q,r}:(0,1]\to(0,\infty)$ with $\varphi(x)\to 0$ as $x\to 0$ such that the following holds: for all $\mathbf{t}\in\mathbb{N}_+^q$ and

$$n\ge (r-1)\|\mathbf{t}\|_\infty+\Lambda_{\mathbf{t}}+1+n\varphi(s/n),$$

if $H$ is an $r$-uniform $n$-vertex $s$-connector, then $H\to \mathbf{t}K_r^r$. This proposes an asymptotic extension of the Alon–Frankl–Lovász Ramsey theorem, with the connector condition playing the role of the complete hypergraph; whether this statement holds for all $r>2$ is the open problem posed by the paper.