Let $F\_1,...,F\_m$ be a partition of the 3-element subsets of $[n]$ into families such that no three subsets in any one family $F\_i$ are all contained in one 4-element subset of $[n]$. What is the minimum value of $m$? 



partitioning the 3-sets of [n]={1,...,n} into families
Tony Huynh's update can be easily generalized to show that
$$n\geq k\left(R(\underbrace{3,3,\dots,3}\_{k-1})-1\right)+3\implies m\geq k$$ and so we get a very weak lower bound on $m$ which at least shows that $m \_{min}\to \infty$.


For an easy upper bound $m\_{min}\le \lfloor\frac{n+1}{2}\rfloor$, which you can see by partitioning the triples $(a,b,c)$ in classes according to $a+b+c\pmod{\lfloor\frac{n+1}{2}\rfloor}$.


