Chvátal's Erdős–Chvátal simplex conjecture

Let $k>r\geq 1$ and $n\geq \frac{r+1}{r}k$. A family $\mathcal{F}\subseteq\binom{[n]}{k}$ is without an $r$-simplex if it contains no $r$-simplex.

Erdős–Chvátal simplex conjecture. Every family $\mathcal{F}\subseteq\binom{[n]}{k}$ without an $r$-simplex contains at most

$$\binom{n-1}{k-1}$$

sets.

This conjecture asserts that the largest uniform families without an $r$-simplex are typically trivial, with the bound attained by the family of all $k$-sets containing a fixed element. Its general status is not specified in the source.