Frankl–Füredi forbidden delta-system conjecture

For fixed $k,\ell,s$, let $f(n,k,\ell,s)$ denote the maximum size of a family of $k$-element sets with no $\Delta(s+1)$-system whose kernel has size $\ell$. Let $\phi(\ell+1,s)$ be the largest size of a family of $(\ell+1)$-element sets containing no $\Delta(s+1)$-system. Frankl–Füredi conjecture.

• If $k\geq 2\ell+1$,

$$f(n,k,\ell,s)\leq (\phi(\ell+1,s)+o(1))\binom{n-\ell-1}{k-\ell-1};$$

• if $k\leq 2\ell$,

$$f(n,k,\ell,s)\leq \frac{\left(\binom{(s+1)(k-\ell)+\ell-1}{k}+o(1)\right)\binom{n}{\ell}}{\binom{(s+1)(k-\ell)+\ell-1}{\ell}}.$$

The two bounds correspond to the two constructions described immediately before the conjecture; the source gives no resolution.