Erdős–Chvátal simplex conjecture

Let $[n]=\{1,\dots,n\}$, let $\binom{[n]}{k}$ denote the family of $k$-subsets of $[n]$, and let a $d$-simplex be a collection of $d+1$ members of $\binom{[n]}{k}$ whose total intersection is empty but every $d$-member subcollection has nonempty intersection. A star is the family of all $k$-subsets containing a fixed element. Erdős–Chvátal simplex conjecture. If $k\geq d+1\geq3$, $n\geq\frac{k(d+1)}{d}$, and $\mathcal{F}\subseteq\binom{[n]}{k}$ contains no $d$-simplex, then

$$|\mathcal{F}|\leq\binom{n-1}{k-1}=\binom{n}{k}-\binom{n-1}{k},$$

with equality only if $\mathcal{F}$ is a star. This generalizes Erdős's triangle conjecture; the source gives no resolution status.