The general arrow-relation conjecture for finite-set traces

Let $[n]=\{1,2,\ldots,n\}$ be the standard $n$-element set and $2^{[n]}$ its powerset. For a family $\mathcal{F}\subset 2^{[n]}$ and $Y\subset [n]$, write $\mathcal{F}_{\mid Y}=\{F\cap Y:F\in\mathcal{F}\}$ for its trace on $Y$. The arrow relation $(n,m)\rightarrow(a,b)$ means that every family $\mathcal{F}\subset 2^{[n]}$ with $|\mathcal{F}|\geq m$ has an $a$-element subset $Y\subset[n]$ with $|\mathcal{F}_{\mid Y}|\geq b$.

The general arrow-relation conjecture. For all integers $n>\ell>0$,

$$\left(n,1+\prod_{0\leq i<\ell} \left\lfloor\frac{n+\ell+i}{\ell}\right\rfloor\right)\rightarrow (\ell+1,3\cdot 2^{\ell-1}+1).$$

The examples preceding the conjecture show that the corresponding lower threshold with the initial $1$ omitted does not force the stated trace size. The conjecture is best possible for $\ell=1$ and $\ell=2$, by the Sauer–Shelah–Vapnik–Chervonenkis theorem and the result attributed to Lovász and proved by the first author; its status for general $\ell$ is left open.